Theorem bj-disjsn01 37128

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9516 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 8417	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2987	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4670	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1542   ≠ wne 2933   ∩ cin 3901  ∅c0 4286  {csn 4581  1_oc1o 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5252

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-nul 4287  df-sn 4582  df-suc 6324  df-1o 8399

This theorem is referenced by: (None)