Theorem bj-0nel1 34389

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 8102	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3044	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5175	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4540	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 326	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3094	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1538   ∈ wcel 2111   ∉ wnel 3091  ∅c0 4243  {csn 4525  1_oc1o 8078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-nel 3092  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-suc 6165  df-1o 8085

This theorem is referenced by: (None)