Theorem bj-0nel1 37307

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 8420	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 2992	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5236	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4577	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 324	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3042	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1547   ∈ wcel 2119   ∉ wnel 3039  ∅c0 4268  {csn 4562  1_oc1o 8395

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-nel 3040  df-v 3434  df-dif 3893  df-un 3895  df-nul 4269  df-sn 4563  df-suc 6323  df-1o 8402

This theorem is referenced by: (None)