Theorem xpcomeng 7937

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 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))

Proof of Theorem xpcomeng

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpeq1 5052	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
2		xpeq2 5053	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
3	1, 2	breq12d 4596	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 × 𝑦) ≈ (𝑦 × 𝑥) ↔ (𝐴 × 𝑦) ≈ (𝑦 × 𝐴)))
4		xpeq2 5053	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
5		xpeq1 5052	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
6	4, 5	breq12d 4596	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 × 𝑦) ≈ (𝑦 × 𝐴) ↔ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)))
7		vex 3176	. . 3 ⊢ 𝑥 ∈ V
8		vex 3176	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	xpcomen 7936	. 2 ⊢ (𝑥 × 𝑦) ≈ (𝑦 × 𝑥)
10	3, 6, 9	vtocl2g 3243	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583   × cxp 5036   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1st 7059  df-2nd 7060  df-en 7842

This theorem is referenced by:  xpsnen2g  7938  xpdom1g  7942  omxpen  7947  xpfir  8067  infxp  8920  infmap2  8923  enrelmap  37311  enrelmapr  37312