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Theorem 0nelelxp 4373
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 4362 . 2  |-  ( C  e.  ( A  X.  B )  <->  E. x E. y ( C  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
2 0nelop 3985 . . . 4  |-  -.  (/)  e.  <. x ,  y >.
3 simpl 102 . . . . 5  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  C  =  <. x ,  y >.
)
43eleq2d 2107 . . . 4  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  ( (/)  e.  C  <->  (/)  e.  <. x ,  y
>. ) )
52, 4mtbiri 600 . . 3  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  -.  (/)  e.  C
)
65exlimivv 1776 . 2  |-  ( E. x E. y ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  -.  (/)  e.  C
)
71, 6sylbi 114 1  |-  ( C  e.  ( A  X.  B )  ->  -.  (/) 
e.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   (/)c0 3224   <.cop 3378    X. cxp 4343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351
This theorem is referenced by:  dmsn0el  4790
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