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Theorem 0nelelxp 4909
 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 4897 . 2
2 0nelop 4448 . . . 4
3 simpl 445 . . . . 5
43eleq2d 2505 . . . 4
52, 4mtbiri 296 . . 3
65exlimivv 1646 . 2
71, 6sylbi 189 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360  wex 1551   wceq 1653   wcel 1726  c0 3630  cop 3819   cxp 4878 This theorem is referenced by:  onxpdisj  4959  dmsn0el  5341 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886
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