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Theorem xpexb 31567
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  X.  B )  e.  _V  <->  ( B  X.  A )  e.  _V )

Proof of Theorem xpexb
StepHypRef Expression
1 cnvxp 5431 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
2 cnvexg 6745 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  `' ( A  X.  B
)  e.  _V )
31, 2syl5eqelr 2550 . 2  |-  ( ( A  X.  B )  e.  _V  ->  ( B  X.  A )  e. 
_V )
4 cnvxp 5431 . . 3  |-  `' ( B  X.  A )  =  ( A  X.  B )
5 cnvexg 6745 . . 3  |-  ( ( B  X.  A )  e.  _V  ->  `' ( B  X.  A
)  e.  _V )
64, 5syl5eqelr 2550 . 2  |-  ( ( B  X.  A )  e.  _V  ->  ( A  X.  B )  e. 
_V )
73, 6impbii 188 1  |-  ( ( A  X.  B )  e.  _V  <->  ( B  X.  A )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1819   _Vcvv 3109    X. cxp 5006   `'ccnv 5007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019
This theorem is referenced by: (None)
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