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Theorem 0nelelxp 4882
Description: A member of a Cartesian 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 4870 . 2  |-  ( C  e.  ( A  X.  B )  <->  E. x E. y ( C  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
2 0nelop 4705 . . . 4  |-  -.  (/)  e.  <. x ,  y >.
3 simpl 463 . . . . 5  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  C  =  <. x ,  y >.
)
43eleq2d 2525 . . . 4  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  ( (/)  e.  C  <->  (/)  e.  <. x ,  y
>. ) )
52, 4mtbiri 309 . . 3  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  -.  (/)  e.  C
)
65exlimivv 1789 . 2  |-  ( E. x E. y ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  -.  (/)  e.  C
)
71, 6sylbi 200 1  |-  ( C  e.  ( A  X.  B )  ->  -.  (/) 
e.  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    = wceq 1455   E.wex 1674    e. wcel 1898   (/)c0 3743   <.cop 3986    X. cxp 4851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-opab 4476  df-xp 4859
This theorem is referenced by:  dmsn0el  5324  onxpdisj  5561
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