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Theorem xpexb 27026
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 5085 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
2 cnvexg 5195 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  `' ( A  X.  B
)  e.  _V )
31, 2syl5eqelr 2343 . 2  |-  ( ( A  X.  B )  e.  _V  ->  ( B  X.  A )  e. 
_V )
4 cnvxp 5085 . . 3  |-  `' ( B  X.  A )  =  ( A  X.  B )
5 cnvexg 5195 . . 3  |-  ( ( B  X.  A )  e.  _V  ->  `' ( B  X.  A
)  e.  _V )
64, 5syl5eqelr 2343 . 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 2763    X. cxp 4659   `'ccnv 4660
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-rel 4676  df-cnv 4677  df-dm 4679  df-rn 4680
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