Theorem bj-disjsn01 37306

Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9522 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 8420	. . 3 ⊢ 1o ≠ ∅
2	1	necomi 2989	. 2 ⊢ ∅ ≠ 1o
3		disjsn2 4651	. 2 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
4	2, 3	ax-mp 5	1 ⊢ ({∅} ∩ {1o}) = ∅

Syntax hints:   = wceq 1547   ≠ wne 2935   ∩ cin 3889  ∅c0 4268  {csn 4562  1_oc1o 8395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-nul 4269  df-sn 4563  df-suc 6323  df-1o 8402

This theorem is referenced by: (None)