Theorem bj-disjsn01 37385

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9548 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 8444	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 3005	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4665	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1554   ≠ wne 2951   ∩ cin 3898  ∅c0 4280  {csn 4576  1_oc1o 8418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-nul 5250

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-nul 4281  df-sn 4577  df-suc 6341  df-1o 8425

This theorem is referenced by: (None)