Theorem bj-disjsn01 37957

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9607 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 8498	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3009	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4669	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

This proof depends on syntax axioms:   = wceq 1569   ≠ wne 2955   ∩ cin 4089  ∅c0 4278  {csn 4584  1_oc1o 8472

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-v 3452  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-sn 4585  df-suc 6368  df-1o 8479

This theorem is used by: (None)