Theorem bj-disjsn01 36933

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9533 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 8429	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2979	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4672	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1540   ≠ wne 2925   ∩ cin 3910  ∅c0 4292  {csn 4585  1_oc1o 8404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5256

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-nul 4293  df-sn 4586  df-suc 6326  df-1o 8411

This theorem is referenced by: (None)