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Theorem xp01disj 8492
Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
Assertion
Ref Expression
xp01disj ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅

Proof of Theorem xp01disj
StepHypRef Expression
1 1n0 8489 . . 3 1o ≠ ∅
21necomi 2989 . 2 ∅ ≠ 1o
3 xpsndisj 6156 . 2 (∅ ≠ 1o → ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅)
42, 3ax-mp 5 1 ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wne 2934  cin 3942  c0 4317  {csn 4623   × cxp 5667  1oc1o 8460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-suc 6364  df-1o 8467
This theorem is referenced by:  endisj  9060
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