Theorem bj-disjsn01 35121

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 35120 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 8300	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2999	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4653	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1541   ≠ wne 2944   ∩ cin 3890  ∅c0 4261  {csn 4566  1_oc1o 8274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-nul 5233

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-ral 3070  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-nul 4262  df-sn 4567  df-suc 6269  df-1o 8281

This theorem is referenced by: (None)