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Theorem xpexb 43702
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 6146 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
2 cnvexg 7908 . . 3 ((𝐴 × 𝐵) ∈ V → (𝐴 × 𝐵) ∈ V)
31, 2eqeltrrid 2830 . 2 ((𝐴 × 𝐵) ∈ V → (𝐵 × 𝐴) ∈ V)
4 cnvxp 6146 . . 3 (𝐵 × 𝐴) = (𝐴 × 𝐵)
5 cnvexg 7908 . . 3 ((𝐵 × 𝐴) ∈ V → (𝐵 × 𝐴) ∈ V)
64, 5eqeltrrid 2830 . 2 ((𝐵 × 𝐴) ∈ V → (𝐴 × 𝐵) ∈ V)
73, 6impbii 208 1 ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098  Vcvv 3466   × cxp 5664  ccnv 5665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677
This theorem is referenced by: (None)
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