Theorem bj-disjsn01 35822

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9596 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 8485	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2996	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4716	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1542   ≠ wne 2941   ∩ cin 3947  ∅c0 4322  {csn 4628  1_oc1o 8456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-nul 4323  df-sn 4629  df-suc 6368  df-1o 8463

This theorem is referenced by: (None)