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Theorem 0nelelxp 4439
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 4428 . 2  |-  ( C  e.  ( A  X.  B )  <->  E. x E. y ( C  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
2 0nelop 4049 . . . 4  |-  -.  (/)  e.  <. x ,  y >.
3 simpl 107 . . . . 5  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  C  =  <. x ,  y >.
)
43eleq2d 2154 . . . 4  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  ( (/)  e.  C  <->  (/)  e.  <. x ,  y
>. ) )
52, 4mtbiri 633 . . 3  |-  ( ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  -.  (/)  e.  C
)
65exlimivv 1821 . 2  |-  ( E. x E. y ( C  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  -.  (/)  e.  C
)
71, 6sylbi 119 1  |-  ( C  e.  ( A  X.  B )  ->  -.  (/) 
e.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1287   E.wex 1424    e. wcel 1436   (/)c0 3275   <.cop 3434    X. cxp 4409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-opab 3875  df-xp 4417
This theorem is referenced by:  dmsn0el  4866
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