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Theorem xpexb 44473
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 6177 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
2 cnvexg 7946 . . 3 ((𝐴 × 𝐵) ∈ V → (𝐴 × 𝐵) ∈ V)
31, 2eqeltrrid 2846 . 2 ((𝐴 × 𝐵) ∈ V → (𝐵 × 𝐴) ∈ V)
4 cnvxp 6177 . . 3 (𝐵 × 𝐴) = (𝐴 × 𝐵)
5 cnvexg 7946 . . 3 ((𝐵 × 𝐴) ∈ V → (𝐵 × 𝐴) ∈ V)
64, 5eqeltrrid 2846 . 2 ((𝐵 × 𝐴) ∈ V → (𝐴 × 𝐵) ∈ V)
73, 6impbii 209 1 ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2108  Vcvv 3480   × cxp 5683  ccnv 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by: (None)
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