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Theorem 0xp 4420
 Description: The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
0xp

Proof of Theorem 0xp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4362 . . 3
2 noel 3228 . . . . . . 7
3 simprl 483 . . . . . . 7
42, 3mto 588 . . . . . 6
54nex 1389 . . . . 5
65nex 1389 . . . 4
7 noel 3228 . . . 4
86, 72false 617 . . 3
91, 8bitri 173 . 2
109eqriv 2037 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1243  wex 1381   wcel 1393  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-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:  res0  4616  xp0  4743  xpeq0r  4746  xpdisj1  4747  xpima1  4767
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