Theorem bj-disjsn01 34879

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 34878 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 8221	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2995	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4628	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1543   ≠ wne 2940   ∩ cin 3865  ∅c0 4237  {csn 4541  1_oc1o 8195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-nul 5199

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-nul 4238  df-sn 4542  df-suc 6219  df-1o 8202

This theorem is referenced by: (None)