Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  0nelelxp Unicode version

Theorem 0nelelxp 5033
 Description: A member of a Cartesian 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 5021 . 2
2 0nelop 4742 . . . 4
3 simpl 457 . . . . 5
43eleq2d 2527 . . . 4
52, 4mtbiri 303 . . 3
65exlimivv 1723 . 2
71, 6sylbi 195 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818   c0 3784  <.cop 4035  X.cxp 5002 This theorem is referenced by:  onxpdisj  5088  dmsn0el  5482 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511  df-xp 5010
 Copyright terms: Public domain W3C validator