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

Theorem 0nelxp 4909
 Description: The empty set is not a member of a cross 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 2961 . . . . . 6
2 vex 2961 . . . . . 6
31, 2opnzi 4436 . . . . 5
4 simpl 445 . . . . . . 7
54eqcomd 2443 . . . . . 6
65necon3ai 2646 . . . . 5
73, 6ax-mp 5 . . . 4
87nex 1565 . . 3
98nex 1565 . 2
10 elxp 4898 . 2
119, 10mtbir 292 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 360  wex 1551   wceq 1653   wcel 1726   wne 2601  c0 3630  cop 3819   cxp 4879 This theorem is referenced by:  onxpdisj  4960  dmsn0  5340  nfunv  5487  mpt2xopx0ov0  6470  reldmtpos  6490  dmtpos  6494  0nnq  8806  adderpq  8838  mulerpq  8839  lterpq  8852  0ncn  9013  structcnvcnv  13485 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pr 4406 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 4270  df-xp 4887
 Copyright terms: Public domain W3C validator