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Theorem xpexb 27658
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 5097 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
2 cnvexg 5208 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  `' ( A  X.  B
)  e.  _V )
31, 2syl5eqelr 2368 . 2  |-  ( ( A  X.  B )  e.  _V  ->  ( B  X.  A )  e. 
_V )
4 cnvxp 5097 . . 3  |-  `' ( B  X.  A )  =  ( A  X.  B )
5 cnvexg 5208 . . 3  |-  ( ( B  X.  A )  e.  _V  ->  `' ( B  X.  A
)  e.  _V )
64, 5syl5eqelr 2368 . 2  |-  ( ( B  X.  A )  e.  _V  ->  ( A  X.  B )  e. 
_V )
73, 6impbii 180 1  |-  ( ( A  X.  B )  e.  _V  <->  ( B  X.  A )  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684   _Vcvv 2788    X. cxp 4687   `'ccnv 4688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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