let X, Y be non empty set ; :: thesis: ( ( for x being object holds
( x in X iff ex N being strict LTLnode over v st x = N ) ) & ( for x being object holds
( x in Y iff ex N being strict LTLnode over v st x = N ) ) implies X = Y )
( ( for x being object holds
( x in X iff ex N being strict LTLnode over v st x = N ) ) & ( for x being object holds
( x in Y iff ex N being strict LTLnode over v st x = N ) ) implies X = Y )

proof

assume that
A13: for x being object holds
( x in X iff ex N being strict LTLnode over v st x = N ) and
A14: for x being object holds
( x in Y iff ex N being strict LTLnode over v st x = N ) ; :: thesis: X = Y
for x being object holds
( x in X iff x in Y )

proof

let x be object ; :: thesis: ( x in X iff x in Y )
( x in X iff ex N being strict LTLnode over v st x = N ) by A13;
hence ( x in X iff x in Y ) by A14; :: thesis: verum

end;

hence X = Y by TARSKI:2; :: thesis: verum

end;

hence ( ( for x being object holds
( x in X iff ex N being strict LTLnode over v st x = N ) ) & ( for x being object holds
( x in Y iff ex N being strict LTLnode over v st x = N ) ) implies X = Y ) ; :: thesis: verum