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Theorem bj-disjsn01 33693
 Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 33692 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 7913 . . 3 1o ≠ ∅
21necomi 3015 . 2 ∅ ≠ 1o
3 disjsn2 4516 . 2 (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
42, 3ax-mp 5 1 ({∅} ∩ {1o}) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1507   ≠ wne 2961   ∩ cin 3824  ∅c0 4173  {csn 4435  1oc1o 7890 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745  ax-nul 5061 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-nul 4174  df-sn 4436  df-suc 6029  df-1o 7897 This theorem is referenced by:  bj-2upln1upl  33789
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