let A, B be finite set ; :: thesis: ( card A < card B implies ex x being set st x in B \ A )

assume card A < card B ; :: thesis: ex x being set st x in B \ A

then not B c= A by NAT_1:43;

then consider x being object such that

A1: x in B and

A2: x nin A ;

take x ; :: thesis: ( x is set & x in B \ A )

thus ( x is set & x in B \ A ) by A1, A2, XBOOLE_0:def 5; :: thesis: verum

assume card A < card B ; :: thesis: ex x being set st x in B \ A

then not B c= A by NAT_1:43;

then consider x being object such that

A1: x in B and

A2: x nin A ;

take x ; :: thesis: ( x is set & x in B \ A )

thus ( x is set & x in B \ A ) by A1, A2, XBOOLE_0:def 5; :: thesis: verum