Theorem bj-disjsn01 37835

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9588 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 8479	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3010	. 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 2956   ∩ cin 3898  ∅c0 4279  {csn 4584  1_oc1o 8453

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-nul 4280  df-sn 4585  df-suc 6361  df-1o 8460

This theorem is used by: (None)