let W, Y be set ; :: thesis: ( ( for x being set holds

( x in W iff ( x c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) ) ) & ( for x being set holds

( x in Y iff ( x c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) ) ) implies W = Y )

assume that

A2: for x being set holds

( x in W iff ( x c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) ) and

A3: for x being set holds

( x in Y iff ( x c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) ) ; :: thesis: W = Y

A4: for x being object st x in Y holds

x in W

x in Y

( x in W iff ( x c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) ) ) & ( for x being set holds

( x in Y iff ( x c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) ) ) implies W = Y )

assume that

A2: for x being set holds

( x in W iff ( x c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) ) and

A3: for x being set holds

( x in Y iff ( x c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) ) ; :: thesis: W = Y

A4: for x being object st x in Y holds

x in W

proof

for x being object st x in W holds
let x be object ; :: thesis: ( x in Y implies x in W )

reconsider xx = x as set by TARSKI:1;

assume x in Y ; :: thesis: x in W

then ( xx c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) by A3;

hence x in W by A2; :: thesis: verum

end;reconsider xx = x as set by TARSKI:1;

assume x in Y ; :: thesis: x in W

then ( xx c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) by A3;

hence x in W by A2; :: thesis: verum

x in Y

proof

hence
W = Y
by A4, TARSKI:2; :: thesis: verum
let x be object ; :: thesis: ( x in W implies x in Y )

reconsider xx = x as set by TARSKI:1;

assume x in W ; :: thesis: x in Y

then ( xx c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) by A2;

hence x in Y by A3; :: thesis: verum

end;reconsider xx = x as set by TARSKI:1;

assume x in W ; :: thesis: x in Y

then ( xx c= Elements N & ex y being Element of Elements N st

( y in X & x = enter (N,y) ) ) by A2;

hence x in Y by A3; :: thesis: verum