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Theorem xp01disj 7843
 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 7842 . . 3 1o ≠ ∅
21necomi 3053 . 2 ∅ ≠ 1o
3 xpsndisj 5798 . 2 (∅ ≠ 1o → ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅)
42, 3ax-mp 5 1 ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1658   ≠ wne 2999   ∩ cin 3797  ∅c0 4144  {csn 4397   × cxp 5340  1oc1o 7819 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-rel 5349  df-cnv 5350  df-suc 5969  df-1o 7826 This theorem is referenced by:  endisj  8316  uncdadom  9308  cdaun  9309  cdaen  9310  cda1dif  9313  pm110.643  9314  cdacomen  9318  cdaassen  9319  xpcdaen  9320  mapcdaen  9321  cdadom1  9323  infcda1  9330
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