Theorem bj-disjsn01 37554

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9571 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 8471	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3010	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4677	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1568   ≠ wne 2956   ∩ cin 3903  ∅c0 4285  {csn 4588  1_oc1o 8445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-nul 4286  df-sn 4589  df-suc 6366  df-1o 8452

This theorem is referenced by: (None)