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Theorem 0nelelxp 4734
 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

Proof of Theorem 0nelelxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4722 . 2
2 0nelop 4272 . . . 4
3 simpl 443 . . . . 5
43eleq2d 2363 . . . 4
52, 4mtbiri 294 . . 3
65exlimivv 1625 . 2
71, 6sylbi 187 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358  wex 1531   wceq 1632   wcel 1696  c0 3468  cop 3656   cxp 4703 This theorem is referenced by:  onxpdisj  4785  dmsn0el  5158 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711
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