Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-disjsn01 Structured version   Visualization version   GIF version

Theorem bj-disjsn01 34407
 Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 34406 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
StepHypRef Expression
1 1n0 8105 . . 3 1o ≠ ∅
21necomi 3041 . 2 ∅ ≠ 1o
3 disjsn2 4608 . 2 (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
42, 3ax-mp 5 1 ({∅} ∩ {1o}) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ≠ wne 2987   ∩ cin 3880  ∅c0 4243  {csn 4525  1oc1o 8081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-nul 5175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-nul 4244  df-sn 4526  df-suc 6166  df-1o 8088 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator