Theorem bj-disjsn01 33693

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 33692 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 7913	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3015	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4516	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1507   ≠ wne 2961   ∩ cin 3824  ∅c0 4173  {csn 4435  1_oc1o 7890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745  ax-nul 5061

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-nul 4174  df-sn 4436  df-suc 6029  df-1o 7897

This theorem is referenced by:  bj-2upln1upl  33789