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Theorem bj-disjsn01 32576
Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 32575 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 7521 . . 3 1𝑜 ≠ ∅
21necomi 2850 . 2 ∅ ≠ 1𝑜
3 disjsn2 4222 . 2 (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
42, 3ax-mp 5 1 ({∅} ∩ {1𝑜}) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wne 2796  cin 3559  c0 3896  {csn 4153  1𝑜c1o 7499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-nul 3897  df-sn 4154  df-suc 5691  df-1o 7506
This theorem is referenced by:  bj-2upln1upl  32651
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