Theorem bj-disjsn01 37621

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9582 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 8481	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3015	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4683	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

This proof depends on syntax axioms:   = wceq 1570   ≠ wne 2961   ∩ cin 3907  ∅c0 4289  {csn 4594  1_oc1o 8455

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 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

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 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-nul 4290  df-sn 4595  df-suc 6373  df-1o 8462

This theorem is used by: (None)