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Theorem 0nelelxp 4718
Description: A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)
Assertion
Ref Expression
0nelelxp  |-  ( C  e.  ( A  X.  B )  ->  -.  (/) 
e.  C )

Proof of Theorem 0nelelxp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4706 . 2  |-  ( C  e.  ( A  X.  B )  <->  E. x E. y ( C  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
2 0nelop 4256 . . . 4  |-  -.  (/)  e.  <. x ,  y >.
3 simpl 443 . . . . 5  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  C  =  <. x ,  y >.
)
43eleq2d 2350 . . . 4  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  ( (/)  e.  C  <->  (/)  e.  <. x ,  y
>. ) )
52, 4mtbiri 294 . . 3  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  -.  (/)  e.  C
)
65exlimivv 1667 . 2  |-  ( E. x E. y ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  -.  (/)  e.  C
)
71, 6sylbi 187 1  |-  ( C  e.  ( A  X.  B )  ->  -.  (/) 
e.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   (/)c0 3455   <.cop 3643    X. cxp 4687
This theorem is referenced by:  onxpdisj  4769  dmsn0el  5142
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695
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