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Theorem bj-0nel1 34766
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 8150 . . . 4 1o ≠ ∅
21nesymi 2991 . . 3 ¬ ∅ = 1o
3 0ex 5175 . . . 4 ∅ ∈ V
43elsn 4531 . . 3 (∅ ∈ {1o} ↔ ∅ = 1o)
52, 4mtbir 326 . 2 ¬ ∅ ∈ {1o}
65nelir 3041 1 ∅ ∉ {1o}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2114  wnel 3038  c0 4211  {csn 4516  1oc1o 8124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-nul 5174
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-nel 3039  df-v 3400  df-dif 3846  df-un 3848  df-nul 4212  df-sn 4517  df-suc 6178  df-1o 8131
This theorem is referenced by: (None)
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