Theorem bj-disjsn01 36323

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9595 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 8483	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2987	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4708	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1533   ≠ wne 2932   ∩ cin 3939  ∅c0 4314  {csn 4620  1_oc1o 8454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5296

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-nul 4315  df-sn 4621  df-suc 6360  df-1o 8461

This theorem is referenced by: (None)