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Theorem xpcomeng 9066
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 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))

Proof of Theorem xpcomeng
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpeq1 5683 . . 3 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
2 xpeq2 5690 . . 3 (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
31, 2breq12d 5154 . 2 (𝑥 = 𝐴 → ((𝑥 × 𝑦) ≈ (𝑦 × 𝑥) ↔ (𝐴 × 𝑦) ≈ (𝑦 × 𝐴)))
4 xpeq2 5690 . . 3 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
5 xpeq1 5683 . . 3 (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
64, 5breq12d 5154 . 2 (𝑦 = 𝐵 → ((𝐴 × 𝑦) ≈ (𝑦 × 𝐴) ↔ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)))
7 vex 3472 . . 3 𝑥 ∈ V
8 vex 3472 . . 3 𝑦 ∈ V
97, 8xpcomen 9065 . 2 (𝑥 × 𝑦) ≈ (𝑦 × 𝑥)
103, 6, 9vtocl2g 3557 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098   class class class wbr 5141   × cxp 5667  cen 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7974  df-2nd 7975  df-en 8942
This theorem is referenced by:  xpsnen2g  9067  xpdom1g  9071  omxpen  9076  xpfir  9268  pwdju1  10187  infxp  10212  infmap2  10215  enrelmap  43321  enrelmapr  43322
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