Theorem xpcomeng 6882

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 4673	. . 3 |- ( x = A -> ( x X. y ) = ( A X. y ) )
2		xpeq2 4674	. . 3 |- ( x = A -> ( y X. x ) = ( y X. A ) )
3	1, 2	breq12d 4042	. 2 |- ( x = A -> ( ( x X. y ) ~~ ( y X. x ) <-> ( A X. y ) ~~ ( y X. A ) ) )
4		xpeq2 4674	. . 3 |- ( y = B -> ( A X. y ) = ( A X. B ) )
5		xpeq1 4673	. . 3 |- ( y = B -> ( y X. A ) = ( B X. A ) )
6	4, 5	breq12d 4042	. 2 |- ( y = B -> ( ( A X. y ) ~~ ( y X. A ) <-> ( A X. B ) ~~ ( B X. A ) ) )
7		vex 2763	. . 3 |- x e. _V
8		vex 2763	. . 3 |- y e. _V
9	7, 8	xpcomen 6881	. 2 |- ( x X. y ) ~~ ( y X. x )
10	3, 6, 9	vtocl2g 2824	1 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   class class class wbr 4029   X. cxp 4657   ~~ cen 6792

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-en 6795

This theorem is referenced by:  xpsnen2g  6883  xpdom1g  6887  hashxp  10897