Theorem xpexb 44898

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 6115	. . 3 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
2		cnvexg 7868	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → ◡(𝐴 × 𝐵) ∈ V)
3	1, 2	eqeltrrid 2842	. 2 ⊢ ((𝐴 × 𝐵) ∈ V → (𝐵 × 𝐴) ∈ V)
4		cnvxp 6115	. . 3 ⊢ ◡(𝐵 × 𝐴) = (𝐴 × 𝐵)
5		cnvexg 7868	. . 3 ⊢ ((𝐵 × 𝐴) ∈ V → ◡(𝐵 × 𝐴) ∈ V)
6	4, 5	eqeltrrid 2842	. 2 ⊢ ((𝐵 × 𝐴) ∈ V → (𝐴 × 𝐵) ∈ V)
7	3, 6	impbii 209	1 ⊢ ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V)

Syntax hints:   ↔ wb 206   ∈ wcel 2114  Vcvv 3430   × cxp 5622  ^◡ccnv 5623

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635

This theorem is referenced by: (None)