Theorem bj-disjsn01 36887

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9625 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 8507	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2985	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4692	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1539   ≠ wne 2931   ∩ cin 3930  ∅c0 4313  {csn 4606  1_oc1o 8480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-nul 5286

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-nul 4314  df-sn 4607  df-suc 6369  df-1o 8487

This theorem is referenced by: (None)