Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xp01disj Structured version   Visualization version   GIF version

Theorem xp01disj 7463
 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 7462 . . 3 1𝑜 ≠ ∅
21necomi 2836 . 2 ∅ ≠ 1𝑜
3 xpsndisj 5476 . 2 (∅ ≠ 1𝑜 → ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅)
42, 3ax-mp 5 1 ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ≠ wne 2780   ∩ cin 3539  ∅c0 3874  {csn 4125   × cxp 5036  1𝑜c1o 7440 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-suc 5646  df-1o 7447 This theorem is referenced by:  endisj  7932  uncdadom  8876  cdaun  8877  cdaen  8878  cda1dif  8881  pm110.643  8882  cdacomen  8886  cdaassen  8887  xpcdaen  8888  mapcdaen  8889  cdadom1  8891  infcda1  8898
 Copyright terms: Public domain W3C validator