Theorem bj-0nel1 36324

Description: The empty set does not belong to {1_o}. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-0nel1	⊢ ∅ ∉ {1o}

Proof of Theorem bj-0nel1

Step	Hyp	Ref	Expression
1		1n0 8483	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 2990	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5297	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4635	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 323	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3041	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1533   ∈ wcel 2098   ∉ wnel 3038  ∅c0 4314  {csn 4620  1_oc1o 8454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5296

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-nel 3039  df-v 3468  df-dif 3943  df-un 3945  df-nul 4315  df-sn 4621  df-suc 6360  df-1o 8461

This theorem is referenced by: (None)