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Theorem xpexb 26825
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 5004 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
2 cnvexg 5114 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  `' ( A  X.  B
)  e.  _V )
31, 2syl5eqelr 2338 . 2  |-  ( ( A  X.  B )  e.  _V  ->  ( B  X.  A )  e. 
_V )
4 cnvxp 5004 . . 3  |-  `' ( B  X.  A )  =  ( A  X.  B )
5 cnvexg 5114 . . 3  |-  ( ( B  X.  A )  e.  _V  ->  `' ( B  X.  A
)  e.  _V )
64, 5syl5eqelr 2338 . 2  |-  ( ( B  X.  A )  e.  _V  ->  ( A  X.  B )  e. 
_V )
73, 6impbii 182 1  |-  ( ( A  X.  B )  e.  _V  <->  ( B  X.  A )  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1621   _Vcvv 2727    X. cxp 4578   `'ccnv 4579
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-cnv 4596  df-dm 4598  df-rn 4599
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