Theorem bj-disjsn01 36965

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9488 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 8398	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2980	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4663	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1541   ≠ wne 2926   ∩ cin 3899  ∅c0 4281  {csn 4574  1_oc1o 8373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-nul 4282  df-sn 4575  df-suc 6308  df-1o 8380

This theorem is referenced by: (None)