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Theorem xp01disj 7521
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 7520 . . 3 1𝑜 ≠ ∅
21necomi 2844 . 2 ∅ ≠ 1𝑜
3 xpsndisj 5516 . 2 (∅ ≠ 1𝑜 → ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅)
42, 3ax-mp 5 1 ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wne 2790  cin 3554  c0 3891  {csn 4148   × cxp 5072  1𝑜c1o 7498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-suc 5688  df-1o 7505
This theorem is referenced by:  endisj  7991  uncdadom  8937  cdaun  8938  cdaen  8939  cda1dif  8942  pm110.643  8943  cdacomen  8947  cdaassen  8948  xpcdaen  8949  mapcdaen  8950  cdadom1  8952  infcda1  8959
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