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Theorem xpexr 6725
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 4577 . . . . . 6  |-  (/)  e.  _V
2 eleq1 2539 . . . . . 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 6717 . . . . 5  |-  ( ( A  X.  B )  e.  C  ->  ran  ( A  X.  B
)  e.  _V )
7 rnxp 5437 . . . . . 6  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
87eleq1d 2536 . . . . 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 2780 . 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 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785    X. cxp 4997   ran crn 5000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by: (None)
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