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Theorem xpexb 38975
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 5586 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
2 cnvexg 7154 . . 3 ((𝐴 × 𝐵) ∈ V → (𝐴 × 𝐵) ∈ V)
31, 2syl5eqelr 2735 . 2 ((𝐴 × 𝐵) ∈ V → (𝐵 × 𝐴) ∈ V)
4 cnvxp 5586 . . 3 (𝐵 × 𝐴) = (𝐴 × 𝐵)
5 cnvexg 7154 . . 3 ((𝐵 × 𝐴) ∈ V → (𝐵 × 𝐴) ∈ V)
64, 5syl5eqelr 2735 . 2 ((𝐵 × 𝐴) ∈ V → (𝐴 × 𝐵) ∈ V)
73, 6impbii 199 1 ((𝐴 × 𝐵) ∈ V ↔ (𝐵 × 𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2030  Vcvv 3231   × cxp 5141  ccnv 5142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by: (None)
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