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Theorem bj-disjsn01 33132
Description: Disjointness of the singletons containing 0 and 1. This is a consequence of bj-disjcsn 33131 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 7663 . . 3 1𝑜 ≠ ∅
21necomi 2918 . 2 ∅ ≠ 1𝑜
3 disjsn2 4322 . 2 (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
42, 3ax-mp 5 1 ({∅} ∩ {1𝑜}) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1564  wne 2864  cin 3647  c0 3991  {csn 4253  1𝑜c1o 7641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-nul 4865
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-v 3274  df-dif 3651  df-un 3653  df-in 3655  df-nul 3992  df-sn 4254  df-suc 5810  df-1o 7648
This theorem is referenced by:  bj-2upln1upl  33207
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