Theorem xpnum

Description: The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013) (Revised by Mario Carneiro, 29-Apr-2015)

	Ref	Expression
Assertion	xpnum	|- ( ( A e. dom card /\ B e. dom card ) -> ( A X. B ) e. dom card )

Proof

Step	Hyp	Ref	Expression
1		isnum2	|- ( A e. dom card <-> E. x e. On x ~~ A )
2		isnum2	|- ( B e. dom card <-> E. y e. On y ~~ B )
3		reeanv	|- ( E. x e. On E. y e. On ( x ~~ A /\ y ~~ B ) <-> ( E. x e. On x ~~ A /\ E. y e. On y ~~ B ) )
4		omcl	|- ( ( x e. On /\ y e. On ) -> ( x .o y ) e. On )
5		omxpen	|- ( ( x e. On /\ y e. On ) -> ( x .o y ) ~~ ( x X. y ) )
6		xpen	|- ( ( x ~~ A /\ y ~~ B ) -> ( x X. y ) ~~ ( A X. B ) )
7		entr	|- ( ( ( x .o y ) ~~ ( x X. y ) /\ ( x X. y ) ~~ ( A X. B ) ) -> ( x .o y ) ~~ ( A X. B ) )
8	5 6 7	syl2an	|- ( ( ( x e. On /\ y e. On ) /\ ( x ~~ A /\ y ~~ B ) ) -> ( x .o y ) ~~ ( A X. B ) )
9		isnumi	|- ( ( ( x .o y ) e. On /\ ( x .o y ) ~~ ( A X. B ) ) -> ( A X. B ) e. dom card )
10	4 8 9	syl2an2r	|- ( ( ( x e. On /\ y e. On ) /\ ( x ~~ A /\ y ~~ B ) ) -> ( A X. B ) e. dom card )
11	10	ex	|- ( ( x e. On /\ y e. On ) -> ( ( x ~~ A /\ y ~~ B ) -> ( A X. B ) e. dom card ) )
12	11	rexlimivv	|- ( E. x e. On E. y e. On ( x ~~ A /\ y ~~ B ) -> ( A X. B ) e. dom card )
13	3 12	sylbir	|- ( ( E. x e. On x ~~ A /\ E. y e. On y ~~ B ) -> ( A X. B ) e. dom card )
14	1 2 13	syl2anb	|- ( ( A e. dom card /\ B e. dom card ) -> ( A X. B ) e. dom card )