Theorem bj-disjsn01 32576

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 32575 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 7521	. . 3 ⊢ 1𝑜 ≠ ∅
2	1	necomi 2850	. 2 ⊢ ∅ ≠ 1𝑜
3		disjsn2 4222	. 2 ⊢ (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1𝑜}) = ∅

Syntax hints:   = wceq 1480   ≠ wne 2796   ∩ cin 3559  ∅c0 3896  {csn 4153  1_𝑜c1o 7499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-nul 4754

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-nul 3897  df-sn 4154  df-suc 5691  df-1o 7506

This theorem is referenced by:  bj-2upln1upl  32651