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Theorem bj-disjsn01 33062
Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 33061 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 7620 . . 3 1𝑜 ≠ ∅
21necomi 2877 . 2 ∅ ≠ 1𝑜
3 disjsn2 4279 . 2 (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
42, 3ax-mp 5 1 ({∅} ∩ {1𝑜}) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wne 2823  cin 3606  c0 3948  {csn 4210  1𝑜c1o 7598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-nul 3949  df-sn 4211  df-suc 5767  df-1o 7605
This theorem is referenced by:  bj-2upln1upl  33137
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