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Metamath Proof Explorer

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 )