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Theorem 0nelxp 4372
 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 2560 . . . . . 6
2 vex 2560 . . . . . 6
31, 2opnzi 3972 . . . . 5
4 simpl 102 . . . . . . 7
54eqcomd 2045 . . . . . 6
65necon3ai 2254 . . . . 5
73, 6ax-mp 7 . . . 4
87nex 1389 . . 3
98nex 1389 . 2
10 elxp 4362 . 2
119, 10mtbir 596 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 97   wceq 1243  wex 1381   wcel 1393   wne 2204  c0 3224  cop 3378   cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351 This theorem is referenced by:  dmsn0  4788  nfunv  4933  reldmtpos  5868  dmtpos  5871  0ncn  6908
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