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Theorem xpexb 39160
 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 5709 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
2 cnvexg 7277 . . 3 ((𝐴 × 𝐵) ∈ V → (𝐴 × 𝐵) ∈ V)
31, 2syl5eqelr 2844 . 2 ((𝐴 × 𝐵) ∈ V → (𝐵 × 𝐴) ∈ V)
4 cnvxp 5709 . . 3 (𝐵 × 𝐴) = (𝐴 × 𝐵)
5 cnvexg 7277 . . 3 ((𝐵 × 𝐴) ∈ V → (𝐵 × 𝐴) ∈ V)
64, 5syl5eqelr 2844 . 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 2139  Vcvv 3340   × cxp 5264  ◡ccnv 5265 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277 This theorem is referenced by: (None)
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