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Theorem bj-disjsn01 36562
Description: Disjointness of the singletons containing 0 and 1. This is a consequence of disjcsn 9629 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 8509 . . 3 1o ≠ ∅
21necomi 2984 . 2 ∅ ≠ 1o
3 disjsn2 4718 . 2 (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
42, 3ax-mp 5 1 ({∅} ∩ {1o}) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wne 2929  cin 3943  c0 4322  {csn 4630  1oc1o 8480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-nul 4323  df-sn 4631  df-suc 6377  df-1o 8487
This theorem is referenced by: (None)
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