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Theorem bj-0nel1 32640
 Description: The empty set does not belong to {1𝑜}. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-0nel1 ∅ ∉ {1𝑜}

Proof of Theorem bj-0nel1
StepHypRef Expression
1 1n0 7535 . . . 4 1𝑜 ≠ ∅
21nesymi 2847 . . 3 ¬ ∅ = 1𝑜
3 0ex 4760 . . . 4 ∅ ∈ V
43elsn 4170 . . 3 (∅ ∈ {1𝑜} ↔ ∅ = 1𝑜)
52, 4mtbir 313 . 2 ¬ ∅ ∈ {1𝑜}
65nelir 2896 1 ∅ ∉ {1𝑜}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∈ wcel 1987   ∉ wnel 2893  ∅c0 3897  {csn 4155  1𝑜c1o 7513 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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759 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 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-v 3192  df-dif 3563  df-un 3565  df-nul 3898  df-sn 4156  df-suc 5698  df-1o 7520 This theorem is referenced by: (None)
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