let W be Universe; :: thesis: ( omega in W implies ex a being Ordinal of W ex M being non empty set st

( a is_cofinal_with omega & M = Rank a & M is_elementary_subsystem_of W ) )

set a = the Ordinal of W;

assume A1: omega in W ; :: thesis: ex a being Ordinal of W ex M being non empty set st

( a is_cofinal_with omega & M = Rank a & M is_elementary_subsystem_of W )

then consider phi being Ordinal-Sequence of W such that

A2: ( phi is increasing & phi is continuous ) and

A3: for a being Ordinal of W

for M being non empty set st phi . a = a & {} <> a & M = Rank a holds

M is_elementary_subsystem_of W by Th33;

consider b being Ordinal of W such that

A4: the Ordinal of W in b and

A5: ( phi . b = b & b is_cofinal_with omega ) by A1, A2, Th29;

reconsider M = Rank b as non empty set by A4, CLASSES1:36;

take b ; :: thesis: ex M being non empty set st

( b is_cofinal_with omega & M = Rank b & M is_elementary_subsystem_of W )

take M ; :: thesis: ( b is_cofinal_with omega & M = Rank b & M is_elementary_subsystem_of W )

thus ( b is_cofinal_with omega & M = Rank b & M is_elementary_subsystem_of W ) by A3, A4, A5; :: thesis: verum

( a is_cofinal_with omega & M = Rank a & M is_elementary_subsystem_of W ) )

set a = the Ordinal of W;

assume A1: omega in W ; :: thesis: ex a being Ordinal of W ex M being non empty set st

( a is_cofinal_with omega & M = Rank a & M is_elementary_subsystem_of W )

then consider phi being Ordinal-Sequence of W such that

A2: ( phi is increasing & phi is continuous ) and

A3: for a being Ordinal of W

for M being non empty set st phi . a = a & {} <> a & M = Rank a holds

M is_elementary_subsystem_of W by Th33;

consider b being Ordinal of W such that

A4: the Ordinal of W in b and

A5: ( phi . b = b & b is_cofinal_with omega ) by A1, A2, Th29;

reconsider M = Rank b as non empty set by A4, CLASSES1:36;

take b ; :: thesis: ex M being non empty set st

( b is_cofinal_with omega & M = Rank b & M is_elementary_subsystem_of W )

take M ; :: thesis: ( b is_cofinal_with omega & M = Rank b & M is_elementary_subsystem_of W )

thus ( b is_cofinal_with omega & M = Rank b & M is_elementary_subsystem_of W ) by A3, A4, A5; :: thesis: verum