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Theorem xpcomeng 9103
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 5703 . . 3 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
2 xpeq2 5710 . . 3 (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
31, 2breq12d 5161 . 2 (𝑥 = 𝐴 → ((𝑥 × 𝑦) ≈ (𝑦 × 𝑥) ↔ (𝐴 × 𝑦) ≈ (𝑦 × 𝐴)))
4 xpeq2 5710 . . 3 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
5 xpeq1 5703 . . 3 (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
64, 5breq12d 5161 . 2 (𝑦 = 𝐵 → ((𝐴 × 𝑦) ≈ (𝑦 × 𝐴) ↔ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)))
7 vex 3482 . . 3 𝑥 ∈ V
8 vex 3482 . . 3 𝑦 ∈ V
97, 8xpcomen 9102 . 2 (𝑥 × 𝑦) ≈ (𝑦 × 𝑥)
103, 6, 9vtocl2g 3574 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106   class class class wbr 5148   × cxp 5687  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1st 8013  df-2nd 8014  df-en 8985
This theorem is referenced by:  xpsnen2g  9104  xpdom1g  9108  omxpen  9113  xpfir  9298  pwdju1  10229  infxp  10252  infmap2  10255  enrelmap  43987  enrelmapr  43988
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