Theorem bj-0nel1 35143

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 8318	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3001	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5231	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4576	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 323	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3052	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1539   ∈ wcel 2106   ∉ wnel 3049  ∅c0 4256  {csn 4561  1_oc1o 8290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-nel 3050  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-suc 6272  df-1o 8297

This theorem is referenced by: (None)