Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### Universal Classes

**by****Bogdan Nowak, and****Grzegorz Bancerek**- Received April 10, 1990
- MML identifier: CLASSES2

- [ Mizar article, MML identifier index ]

environ vocabulary CARD_1, ORDINAL1, FUNCT_1, CLASSES1, TARSKI, BOOLE, RELAT_1, ORDINAL2, CARD_3, FUNCT_2, PROB_1, RLVECT_1, ZF_LANG, ZFMISC_1, SETFAM_1, FINSET_1, CLASSES2, HAHNBAN; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NAT_1, WELLORD2, FINSET_1, SETFAM_1, RELAT_1, FUNCT_1, FUNCT_2, ORDINAL1, ORDINAL2, CARD_1, CLASSES1, PROB_1, CARD_3; constructors WELLORD2, PROB_1, CLASSES1, CARD_3, XCMPLX_0, MEMBERED, PARTFUN1, RELAT_2, XBOOLE_0; clusters ORDINAL1, CARD_1, CLASSES1, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1, FUNCT_2, PARTFUN1, XBOOLE_0, ORDINAL2; requirements SUBSET, BOOLE; begin reserve m for Cardinal, n for Nat, A,B,C for Ordinal, x,y,z,X,Y,Z,W for set, f for Function; definition cluster being_Tarski-Class -> subset-closed set; end; definition let X be set; cluster Tarski-Class X -> being_Tarski-Class; end; theorem :: CLASSES2:1 W is subset-closed & X in W implies not X,W are_equipotent & Card X <` Card W; canceled; theorem :: CLASSES2:3 W is_Tarski-Class & x in W & y in W implies {x} in W & {x,y} in W; theorem :: CLASSES2:4 W is_Tarski-Class & x in W & y in W implies [x,y] in W; theorem :: CLASSES2:5 W is_Tarski-Class & X in W implies Tarski-Class X c= W; scheme TC {P[set]}: for X holds P[Tarski-Class X] provided for X st X is_Tarski-Class holds P[X]; theorem :: CLASSES2:6 W is_Tarski-Class & A in W implies succ A in W & A c= W; theorem :: CLASSES2:7 A in Tarski-Class W implies succ A in Tarski-Class W & A c= Tarski-Class W; theorem :: CLASSES2:8 W is subset-closed & X is epsilon-transitive & X in W implies X c= W; theorem :: CLASSES2:9 X is epsilon-transitive & X in Tarski-Class W implies X c= Tarski-Class W; theorem :: CLASSES2:10 W is_Tarski-Class implies On W = Card W; theorem :: CLASSES2:11 On Tarski-Class W = Card Tarski-Class W; theorem :: CLASSES2:12 W is_Tarski-Class & X in W implies Card X in W; theorem :: CLASSES2:13 X in Tarski-Class W implies Card X in Tarski-Class W; theorem :: CLASSES2:14 W is_Tarski-Class & x in Card W implies x in W; theorem :: CLASSES2:15 x in Card Tarski-Class W implies x in Tarski-Class W; theorem :: CLASSES2:16 W is_Tarski-Class & m <` Card W implies m in W; theorem :: CLASSES2:17 m <` Card Tarski-Class W implies m in Tarski-Class W; theorem :: CLASSES2:18 W is_Tarski-Class & m in W implies m c= W; theorem :: CLASSES2:19 m in Tarski-Class W implies m c= Tarski-Class W; theorem :: CLASSES2:20 W is_Tarski-Class implies Card W is_limit_ordinal; theorem :: CLASSES2:21 W is_Tarski-Class & W <> {} implies Card W <> 0 & Card W <> {} & Card W is_limit_ordinal; theorem :: CLASSES2:22 Card Tarski-Class W <> 0 & Card Tarski-Class W <> {} & Card Tarski-Class W is_limit_ordinal; reserve f,g for Function, L,L1 for T-Sequence, F for Cardinal-Function; theorem :: CLASSES2:23 W is_Tarski-Class & (X in W & W is epsilon-transitive or X in W & X c= W or Card X <` Card W & X c= W) implies Funcs(X,W) c= W; theorem :: CLASSES2:24 X in Tarski-Class W & W is epsilon-transitive or X in Tarski-Class W & X c= Tarski-Class W or Card X <` Card Tarski-Class W & X c= Tarski-Class W implies Funcs(X,Tarski-Class W) c= Tarski-Class W; theorem :: CLASSES2:25 dom L is_limit_ordinal & (for A st A in dom L holds L.A = Rank A) implies Rank dom L = Union L; theorem :: CLASSES2:26 W is_Tarski-Class & A in On W implies Card Rank A <` Card W & Rank A in W; theorem :: CLASSES2:27 A in On Tarski-Class W implies Card Rank A <` Card Tarski-Class W & Rank A in Tarski-Class W; theorem :: CLASSES2:28 W is_Tarski-Class implies Rank Card W c= W; theorem :: CLASSES2:29 Rank Card Tarski-Class W c= Tarski-Class W; theorem :: CLASSES2:30 W is_Tarski-Class & W is epsilon-transitive & X in W implies the_rank_of X in W; theorem :: CLASSES2:31 W is_Tarski-Class & W is epsilon-transitive implies W c= Rank Card W; theorem :: CLASSES2:32 W is_Tarski-Class & W is epsilon-transitive implies Rank Card W = W; theorem :: CLASSES2:33 W is_Tarski-Class & A in On W implies Card Rank A <=` Card W; theorem :: CLASSES2:34 A in On Tarski-Class W implies Card Rank A <=` Card Tarski-Class W; theorem :: CLASSES2:35 W is_Tarski-Class implies Card W = Card Rank Card W; theorem :: CLASSES2:36 Card Tarski-Class W = Card Rank Card Tarski-Class W; theorem :: CLASSES2:37 W is_Tarski-Class & X c= Rank Card W implies X,Rank Card W are_equipotent or X in Rank Card W; theorem :: CLASSES2:38 X c= Rank Card Tarski-Class W implies X,Rank Card Tarski-Class W are_equipotent or X in Rank Card Tarski-Class W; theorem :: CLASSES2:39 W is_Tarski-Class implies Rank Card W is_Tarski-Class; theorem :: CLASSES2:40 Rank Card Tarski-Class W is_Tarski-Class; theorem :: CLASSES2:41 X is epsilon-transitive & A in the_rank_of X implies ex Y st Y in X & the_rank_of Y = A; theorem :: CLASSES2:42 X is epsilon-transitive implies Card the_rank_of X <=` Card X; theorem :: CLASSES2:43 W is_Tarski-Class & X is epsilon-transitive & X in W implies X in Rank Card W; theorem :: CLASSES2:44 X is epsilon-transitive & X in Tarski-Class W implies X in Rank Card Tarski-Class W; theorem :: CLASSES2:45 W is epsilon-transitive implies Rank Card Tarski-Class W is_Tarski-Class_of W; theorem :: CLASSES2:46 W is epsilon-transitive implies Rank Card Tarski-Class W = Tarski-Class W; definition let IT be set; attr IT is universal means :: CLASSES2:def 1 IT is epsilon-transitive & IT is_Tarski-Class; end; definition cluster universal -> epsilon-transitive being_Tarski-Class set; cluster epsilon-transitive being_Tarski-Class -> universal set; end; definition cluster universal non empty set; end; definition mode Universe is universal non empty set; end; reserve U1,U2,U3,Universum for Universe; canceled 3; theorem :: CLASSES2:50 On Universum is Ordinal; theorem :: CLASSES2:51 X is epsilon-transitive implies Tarski-Class X is universal; theorem :: CLASSES2:52 Tarski-Class Universum is Universe; definition let Universum; cluster On Universum -> ordinal; cluster Tarski-Class Universum -> universal; end; theorem :: CLASSES2:53 Tarski-Class A is universal; definition let A; cluster Tarski-Class A -> universal; end; theorem :: CLASSES2:54 Universum = Rank On Universum; theorem :: CLASSES2:55 On Universum <> {} & On Universum is_limit_ordinal; theorem :: CLASSES2:56 U1 in U2 or U1 = U2 or U2 in U1; theorem :: CLASSES2:57 U1 c= U2 or U2 in U1; theorem :: CLASSES2:58 U1,U2 are_c=-comparable; theorem :: CLASSES2:59 U1 in U2 & U2 in U3 implies U1 in U3; canceled; theorem :: CLASSES2:61 U1 \/ U2 is Universe & U1 /\ U2 is Universe; theorem :: CLASSES2:62 {} in Universum; theorem :: CLASSES2:63 x in Universum implies {x} in Universum; theorem :: CLASSES2:64 x in Universum & y in Universum implies {x,y} in Universum & [x,y] in Universum; theorem :: CLASSES2:65 X in Universum implies bool X in Universum & union X in Universum & meet X in Universum; theorem :: CLASSES2:66 X in Universum & Y in Universum implies X \/ Y in Universum & X /\ Y in Universum & X \ Y in Universum & X \+\ Y in Universum; theorem :: CLASSES2:67 X in Universum & Y in Universum implies [:X,Y:] in Universum & Funcs(X,Y) in Universum; reserve u,v for Element of Universum; definition let U1; cluster non empty Element of U1; end; definition let Universum,u; redefine func {u} -> Element of Universum; func bool u -> non empty Element of Universum; func union u -> Element of Universum; func meet u -> Element of Universum; let v; func {u,v} -> Element of Universum; func [u,v] -> Element of Universum; func u \/ v -> Element of Universum; func u /\ v -> Element of Universum; func u \ v -> Element of Universum; func u \+\ v -> Element of Universum; func [:u,v:] -> Element of Universum; func Funcs(u,v) -> Element of Universum; end; definition func FinSETS -> Universe equals :: CLASSES2:def 2 Tarski-Class {}; end; canceled; theorem :: CLASSES2:69 Card Rank omega = Card omega; theorem :: CLASSES2:70 Rank omega is_Tarski-Class; theorem :: CLASSES2:71 FinSETS = Rank omega; definition func SETS -> Universe equals :: CLASSES2:def 3 Tarski-Class FinSETS; end; definition let X be set; cluster the_transitive-closure_of X -> epsilon-transitive; end; definition let X be epsilon-transitive set; cluster Tarski-Class X -> epsilon-transitive; end; definition let A be Ordinal; cluster Rank A -> epsilon-transitive; end; definition let X be set; func Universe_closure X -> Universe means :: CLASSES2:def 4 X c= it & for Y being Universe st X c= Y holds it c= Y; end; definition mode FinSet is Element of FinSETS; mode Set is Element of SETS; let A; func UNIVERSE A means :: CLASSES2:def 5 ex L st it = last L & dom L = succ A & L.{} = FinSETS & (for C st succ C in succ A holds L.succ C = Tarski-Class(L.C)) & for C st C in succ A & C <> {} & C is_limit_ordinal holds L.C = Universe_closure Union(L|C); end; definition let A; cluster UNIVERSE A -> universal non empty; end; canceled 3; theorem :: CLASSES2:75 UNIVERSE {} = FinSETS; theorem :: CLASSES2:76 UNIVERSE succ A = Tarski-Class UNIVERSE A; theorem :: CLASSES2:77 UNIVERSE one = SETS; theorem :: CLASSES2:78 A <> {} & A is_limit_ordinal & dom L = A & (for B st B in A holds L.B = UNIVERSE B) implies UNIVERSE A = Universe_closure Union L; theorem :: CLASSES2:79 FinSETS c= Universum & Tarski-Class {} c= Universum & UNIVERSE {} c= Universum; theorem :: CLASSES2:80 A in B iff UNIVERSE A in UNIVERSE B; theorem :: CLASSES2:81 UNIVERSE A = UNIVERSE B implies A = B; theorem :: CLASSES2:82 A c= B iff UNIVERSE A c= UNIVERSE B;

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