MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpexr Structured version   Unicode version

Theorem xpexr 6713
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 4569 . . . . . 6  |-  (/)  e.  _V
2 eleq1 2526 . . . . . 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 6705 . . . . 5  |-  ( ( A  X.  B )  e.  C  ->  ran  ( A  X.  B
)  e.  _V )
7 rnxp 5422 . . . . . 6  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
87eleq1d 2523 . . . . 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 2767 . 2  |-  ( ( A  X.  B )  e.  C  ->  ( -.  A  e.  _V  ->  B  e.  _V )
)
1211orrd 376 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 366    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106   (/)c0 3783    X. cxp 4986   ran crn 4989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator