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Theorem 0nelxp 5032
Description: The empty set is not a member of a Cartesian product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
0nelxp

Proof of Theorem 0nelxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . . . 6
2 vex 3112 . . . . . 6
31, 2opnzi 4724 . . . . 5
4 simpl 457 . . . . . . 7
54eqcomd 2465 . . . . . 6
65necon3ai 2685 . . . . 5
73, 6ax-mp 5 . . . 4
87nex 1627 . . 3
98nex 1627 . 2
10 elxp 5021 . 2
119, 10mtbir 299 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652   c0 3784  <.cop 4035  X.cxp 5002
This theorem is referenced by:  onxpdisj  5088  dmsn0  5480  nfunv  5624  mpt2xopx0ov0  6963  reldmtpos  6982  dmtpos  6986  0nnq  9323  adderpq  9355  mulerpq  9356  lterpq  9369  0ncn  9531  structcnvcnv  14643  msrrcl  28903
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
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