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Theorem xpcomeng 9095
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 5696 . . 3 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
2 xpeq2 5703 . . 3 (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
31, 2breq12d 5165 . 2 (𝑥 = 𝐴 → ((𝑥 × 𝑦) ≈ (𝑦 × 𝑥) ↔ (𝐴 × 𝑦) ≈ (𝑦 × 𝐴)))
4 xpeq2 5703 . . 3 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
5 xpeq1 5696 . . 3 (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
64, 5breq12d 5165 . 2 (𝑦 = 𝐵 → ((𝐴 × 𝑦) ≈ (𝑦 × 𝐴) ↔ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)))
7 vex 3477 . . 3 𝑥 ∈ V
8 vex 3477 . . 3 𝑦 ∈ V
97, 8xpcomen 9094 . 2 (𝑥 × 𝑦) ≈ (𝑦 × 𝑥)
103, 6, 9vtocl2g 3562 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098   class class class wbr 5152   × cxp 5680  cen 8967
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-1st 7999  df-2nd 8000  df-en 8971
This theorem is referenced by:  xpsnen2g  9096  xpdom1g  9100  omxpen  9105  xpfir  9297  pwdju1  10221  infxp  10246  infmap2  10249  enrelmap  43458  enrelmapr  43459
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