Theorem xp01disj 8415

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 8412	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2983	. 2 ⊢ ∅ ≠ 1o
3		xpsndisj 6118	. 2 ⊢ (∅ ≠ 1o → ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅

Syntax hints:   = wceq 1541   ≠ wne 2929   ∩ cin 3897  ∅c0 4282  {csn 4577   × cxp 5619  1_oc1o 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-opab 5158  df-xp 5627  df-rel 5628  df-suc 6320  df-1o 8394

This theorem is referenced by:  endisj  8988