Theorem xp01disj 8508

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

Step	Hyp	Ref	Expression
1		1n0 8505	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2987	. 2 ⊢ ∅ ≠ 1o
3		xpsndisj 6157	. 2 ⊢ (∅ ≠ 1o → ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅

Syntax hints:   = wceq 1540   ≠ wne 2933   ∩ cin 3930  ∅c0 4313  {csn 4606   × cxp 5657  1_oc1o 8478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-cnv 5667  df-suc 6363  df-1o 8485

This theorem is referenced by:  endisj  9077