Theorem bj-disjsn01 37007

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9503 but the present proof does not use regularity. (Contributed by BJ, 4-Apr-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-disjsn01	⊢ ({∅} ∩ {1o}) = ∅

Proof of Theorem bj-disjsn01

Step	Hyp	Ref	Expression
1		1n0 8412	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2984	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4666	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1541   ≠ wne 2930   ∩ cin 3898  ∅c0 4284  {csn 4577  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-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-ral 3050  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-nul 4285  df-sn 4578  df-suc 6320  df-1o 8394

This theorem is referenced by: (None)