Theorem xpexb 43203

Description: A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)

Assertion

Ref	Expression
xpexb	⊢ ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V)

Proof of Theorem xpexb

Step	Hyp	Ref	Expression
1		cnvxp 6156	. . 3 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
2		cnvexg 7914	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → ◡(𝐴 × 𝐵) ∈ V)
3	1, 2	eqeltrrid 2838	. 2 ⊢ ((𝐴 × 𝐵) ∈ V → (𝐵 × 𝐴) ∈ V)
4		cnvxp 6156	. . 3 ⊢ ◡(𝐵 × 𝐴) = (𝐴 × 𝐵)
5		cnvexg 7914	. . 3 ⊢ ((𝐵 × 𝐴) ∈ V → ◡(𝐵 × 𝐴) ∈ V)
6	4, 5	eqeltrrid 2838	. 2 ⊢ ((𝐵 × 𝐴) ∈ V → (𝐴 × 𝐵) ∈ V)
7	3, 6	impbii 208	1 ⊢ ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V)

Syntax hints:   ↔ wb 205   ∈ wcel 2106  Vcvv 3474   × cxp 5674  ^◡ccnv 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687

This theorem is referenced by: (None)