Theorem bj-disjsn01 36947

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9648 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 8531	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2994	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4718	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1538   ≠ wne 2939   ∩ cin 3963  ∅c0 4340  {csn 4632  1_oc1o 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5313

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-v 3481  df-dif 3967  df-un 3969  df-in 3971  df-nul 4341  df-sn 4633  df-suc 6395  df-1o 8511

This theorem is referenced by: (None)