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Theorem xpexr 6537
Description: If a Cartesian product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)
Assertion
Ref Expression
xpexr  |-  ( ( A  X.  B )  e.  C  ->  ( A  e.  _V  \/  B  e.  _V )
)

Proof of Theorem xpexr
StepHypRef Expression
1 0ex 4441 . . . . . 6  |-  (/)  e.  _V
2 eleq1 2503 . . . . . 6  |-  ( A  =  (/)  ->  ( A  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 233 . . . . 5  |-  ( A  =  (/)  ->  A  e. 
_V )
43pm2.24d 143 . . . 4  |-  ( A  =  (/)  ->  ( -.  A  e.  _V  ->  B  e.  _V ) )
54a1d 25 . . 3  |-  ( A  =  (/)  ->  ( ( A  X.  B )  e.  C  ->  ( -.  A  e.  _V  ->  B  e.  _V )
) )
6 rnexg 6529 . . . . 5  |-  ( ( A  X.  B )  e.  C  ->  ran  ( A  X.  B
)  e.  _V )
7 rnxp 5287 . . . . . 6  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
87eleq1d 2509 . . . . 5  |-  ( A  =/=  (/)  ->  ( ran  ( A  X.  B
)  e.  _V  <->  B  e.  _V ) )
96, 8syl5ib 219 . . . 4  |-  ( A  =/=  (/)  ->  ( ( A  X.  B )  e.  C  ->  B  e.  _V ) )
109a1dd 46 . . 3  |-  ( A  =/=  (/)  ->  ( ( A  X.  B )  e.  C  ->  ( -.  A  e.  _V  ->  B  e.  _V ) ) )
115, 10pm2.61ine 2709 . 2  |-  ( ( A  X.  B )  e.  C  ->  ( -.  A  e.  _V  ->  B  e.  _V )
)
1211orrd 378 1  |-  ( ( A  X.  B )  e.  C  ->  ( A  e.  _V  \/  B  e.  _V )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    = wceq 1369    e. wcel 1756    =/= wne 2620   _Vcvv 2991   (/)c0 3656    X. cxp 4857   ran crn 4860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-xp 4865  df-rel 4866  df-cnv 4867  df-dm 4869  df-rn 4870
This theorem is referenced by: (None)
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