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Theorem xpexb 39438
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
StepHypRef Expression
1 cnvxp 5768 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
2 cnvexg 7347 . . 3 ((𝐴 × 𝐵) ∈ V → (𝐴 × 𝐵) ∈ V)
31, 2syl5eqelr 2883 . 2 ((𝐴 × 𝐵) ∈ V → (𝐵 × 𝐴) ∈ V)
4 cnvxp 5768 . . 3 (𝐵 × 𝐴) = (𝐴 × 𝐵)
5 cnvexg 7347 . . 3 ((𝐵 × 𝐴) ∈ V → (𝐵 × 𝐴) ∈ V)
64, 5syl5eqelr 2883 . 2 ((𝐵 × 𝐴) ∈ V → (𝐴 × 𝐵) ∈ V)
73, 6impbii 201 1 ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wcel 2157  Vcvv 3385   × cxp 5310  ccnv 5311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rn 5323
This theorem is referenced by: (None)
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