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Theorem xpnz 5431
Description: The Cartesian product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.)
Assertion
Ref Expression
xpnz

Proof of Theorem xpnz
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3794 . . . . 5
2 n0 3794 . . . . 5
31, 2anbi12i 697 . . . 4
4 eeanv 1988 . . . 4
53, 4bitr4i 252 . . 3
6 opex 4716 . . . . . 6
7 eleq1 2529 . . . . . . 7
8 opelxp 5034 . . . . . . 7
97, 8syl6bb 261 . . . . . 6
106, 9spcev 3201 . . . . 5
11 n0 3794 . . . . 5
1210, 11sylibr 212 . . . 4
1312exlimivv 1723 . . 3
145, 13sylbi 195 . 2
15 xpeq1 5018 . . . . 5
16 0xp 5085 . . . . 5
1715, 16syl6eq 2514 . . . 4
1817necon3i 2697 . . 3
19 xpeq2 5019 . . . . 5
20 xp0 5430 . . . . 5
2119, 20syl6eq 2514 . . . 4
2221necon3i 2697 . . 3
2318, 22jca 532 . 2
2414, 23impbii 188 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652   c0 3784  <.cop 4035  X.cxp 5002
This theorem is referenced by:  xpeq0  5432  ssxpb  5446  xp11  5447  unixpid  5547  xpexr2  6741  frxp  6910  xpfir  7762  axcc2lem  8837  axdc4lem  8856  mamufacex  18891  txindis  20135  bj-xpnzex  34516  bj-1upln0  34567  bj-2upln1upl  34582  dibn0  36880
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-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  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-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012
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