Theorem xpcomeng 6302

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 4359	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
2		xpeq2 4360	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
3	1, 2	breq12d 3777	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 × 𝑦) ≈ (𝑦 × 𝑥) ↔ (𝐴 × 𝑦) ≈ (𝑦 × 𝐴)))
4		xpeq2 4360	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
5		xpeq1 4359	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
6	4, 5	breq12d 3777	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 × 𝑦) ≈ (𝑦 × 𝐴) ↔ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)))
7		vex 2560	. . 3 ⊢ 𝑥 ∈ V
8		vex 2560	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	xpcomen 6301	. 2 ⊢ (𝑥 × 𝑦) ≈ (𝑦 × 𝑥)
10	3, 6, 9	vtocl2g 2617	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393   class class class wbr 3764   × cxp 4343   ≈ cen 6219

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-1st 5767  df-2nd 5768  df-en 6222

This theorem is referenced by:  xpsnen2g  6303  xpdom1g  6307