Theorem bj-disjsn01 38043

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9631 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 8523	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3013	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4673	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

This proof depends on syntax axioms:   = wceq 1570   ≠ wne 2959   ∩ cin 4092  ∅c0 4281  {csn 4588  1_oc1o 8497

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-v 3456  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-sn 4589  df-suc 6386  df-1o 8504

This theorem is used by: (None)