Theorem xpcomeng 6896

Description: Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)

Assertion

Ref	Expression
xpcomeng	|- ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) )

Proof of Theorem xpcomeng

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpeq1 4678	. . 3 |- ( x = A -> ( x X. y ) = ( A X. y ) )
2		xpeq2 4679	. . 3 |- ( x = A -> ( y X. x ) = ( y X. A ) )
3	1, 2	breq12d 4047	. 2 |- ( x = A -> ( ( x X. y ) ~~ ( y X. x ) <-> ( A X. y ) ~~ ( y X. A ) ) )
4		xpeq2 4679	. . 3 |- ( y = B -> ( A X. y ) = ( A X. B ) )
5		xpeq1 4678	. . 3 |- ( y = B -> ( y X. A ) = ( B X. A ) )
6	4, 5	breq12d 4047	. 2 |- ( y = B -> ( ( A X. y ) ~~ ( y X. A ) <-> ( A X. B ) ~~ ( B X. A ) ) )
7		vex 2766	. . 3 |- x e. _V
8		vex 2766	. . 3 |- y e. _V
9	7, 8	xpcomen 6895	. 2 |- ( x X. y ) ~~ ( y X. x )
10	3, 6, 9	vtocl2g 2828	1 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   class class class wbr 4034   X. cxp 4662   ~~ cen 6806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-1st 6207  df-2nd 6208  df-en 6809

This theorem is referenced by:  xpsnen2g  6897  xpdom1g  6901  hashxp  10935