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Theorem xpexb 27572
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 5281 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
2 cnvexg 5396 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  `' ( A  X.  B
)  e.  _V )
31, 2syl5eqelr 2520 . 2  |-  ( ( A  X.  B )  e.  _V  ->  ( B  X.  A )  e. 
_V )
4 cnvxp 5281 . . 3  |-  `' ( B  X.  A )  =  ( A  X.  B )
5 cnvexg 5396 . . 3  |-  ( ( B  X.  A )  e.  _V  ->  `' ( B  X.  A
)  e.  _V )
64, 5syl5eqelr 2520 . 2  |-  ( ( B  X.  A )  e.  _V  ->  ( A  X.  B )  e. 
_V )
73, 6impbii 181 1  |-  ( ( A  X.  B )  e.  _V  <->  ( B  X.  A )  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1725   _Vcvv 2948    X. cxp 4867   `'ccnv 4868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877  df-dm 4879  df-rn 4880
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