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Theorem bj-disjsn01 32130
 Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 32129 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 7462 . . 3 1𝑜 ≠ ∅
21necomi 2836 . 2 ∅ ≠ 1𝑜
3 disjsn2 4193 . 2 (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
42, 3ax-mp 5 1 ({∅} ∩ {1𝑜}) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ≠ wne 2780   ∩ cin 3539  ∅c0 3874  {csn 4125  1𝑜c1o 7440 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-nul 3875  df-sn 4126  df-suc 5646  df-1o 7447 This theorem is referenced by:  bj-2upln1upl  32205
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