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Theorem 0nelelxp 4538
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 4526 . 2  |-  ( C  e.  ( A  X.  B )  <->  E. x E. y ( C  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
2 0nelop 4140 . . . 4  |-  -.  (/)  e.  <. x ,  y >.
3 simpl 108 . . . . 5  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  C  =  <. x ,  y >.
)
43eleq2d 2187 . . . 4  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  ( (/)  e.  C  <->  (/)  e.  <. x ,  y
>. ) )
52, 4mtbiri 649 . . 3  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  -.  (/)  e.  C
)
65exlimivv 1852 . 2  |-  ( E. x E. y ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  -.  (/)  e.  C
)
71, 6sylbi 120 1  |-  ( C  e.  ( A  X.  B )  ->  -.  (/) 
e.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1316   E.wex 1453    e. wcel 1465   (/)c0 3333   <.cop 3500    X. cxp 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-xp 4515
This theorem is referenced by:  dmsn0el  4978
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