Theorem xpcomeng 6730

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 4561	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
2		xpeq2 4562	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
3	1, 2	breq12d 3950	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 × 𝑦) ≈ (𝑦 × 𝑥) ↔ (𝐴 × 𝑦) ≈ (𝑦 × 𝐴)))
4		xpeq2 4562	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
5		xpeq1 4561	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
6	4, 5	breq12d 3950	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 × 𝑦) ≈ (𝑦 × 𝐴) ↔ (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)))
7		vex 2692	. . 3 ⊢ 𝑥 ∈ V
8		vex 2692	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	xpcomen 6729	. 2 ⊢ (𝑥 × 𝑦) ≈ (𝑦 × 𝑥)
10	3, 6, 9	vtocl2g 2753	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481   class class class wbr 3937   × cxp 4545   ≈ cen 6640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-en 6643

This theorem is referenced by:  xpsnen2g  6731  xpdom1g  6735  hashxp  10604