let X1, X2 be Subset of X; :: thesis: ( ( for x being set holds
( x in X1 iff ( x in X & x is finite ) ) ) & ( for x being set holds
( x in X2 iff ( x in X & x is finite ) ) ) implies X1 = X2 )
assume A1: ( ( for x being set holds
( x in X1 iff ( x in X & x is finite ) ) ) & ( for x being set holds
( x in X2 iff ( x in X & x is finite ) ) ) & not X1 = X2 ) ; :: thesis: contradiction
then consider x being set such that
A2: ( ( x in X1 & not x in X2 ) or ( x in X2 & not x in X1 ) ) by TARSKI:1;
( x in X2 iff ( not x in X or not x is finite ) ) by A1, A2;
hence contradiction by A1; :: thesis: verum