Theorem xpcomeng 4440

Description: Commutative law for equinumerosity of cross product. Proposition 4.22(d) of [Mendelson] p. 254.

Assertion

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

Proof of Theorem xpcomeng

Step	Hyp	Ref	Expression
1		xpeq1 3200	. . 3 |- (x = A -> (x X. y) = (A X. y))
2		xpeq2 3201	. . 3 |- (x = A -> (y X. x) = (y X. A))
3	1, 2	breq12d 2631	. 2 |- (x = A -> ((x X. y) ~~ (y X. x) <-> (A X. y) ~~ (y X. A)))
4		xpeq2 3201	. . 3 |- (y = B -> (A X. y) = (A X. B))
5		xpeq1 3200	. . 3 |- (y = B -> (y X. A) = (B X. A))
6	4, 5	breq12d 2631	. 2 |- (y = B -> ((A X. y) ~~ (y X. A) <-> (A X. B) ~~ (B X. A)))
7		visset 1813	. . 3 |- x e. V
8		visset 1813	. . 3 |- y e. V
9	7, 8	xpcomen 4439	. 2 |- (x X. y) ~~ (y X. x)
10	3, 6, 9	vtocl2g 1850	1 |- ((A e. C /\ B e. D) -> (A X. B) ~~ (B X. A))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958   class class class wbr 2619   X. cxp 3168   ~~ cen 4364

This theorem is referenced by:  xpdom1 4443

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-en 4368

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