Theorem xpsnen2g

Description: A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014)

	Ref	Expression
Assertion	xpsnen2g	|- ( ( A e. V /\ B e. W ) -> ( { A } X. B ) ~~ B )

Proof

Step	Hyp	Ref	Expression
1		snex	|- { A } e. _V
2		xpcomeng	|- ( ( { A } e. _V /\ B e. W ) -> ( { A } X. B ) ~~ ( B X. { A } ) )
3	1 2	mpan	|- ( B e. W -> ( { A } X. B ) ~~ ( B X. { A } ) )
4		xpsneng	|- ( ( B e. W /\ A e. V ) -> ( B X. { A } ) ~~ B )
5	4	ancoms	|- ( ( A e. V /\ B e. W ) -> ( B X. { A } ) ~~ B )
6		entr	|- ( ( ( { A } X. B ) ~~ ( B X. { A } ) /\ ( B X. { A } ) ~~ B ) -> ( { A } X. B ) ~~ B )
7	3 5 6	syl2an2	|- ( ( A e. V /\ B e. W ) -> ( { A } X. B ) ~~ B )