Theorem bj-0nel1 37386

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 8444	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3008	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5251	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4591	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 325	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3058	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1554   ∈ wcel 2136   ∉ wnel 3055  ∅c0 4280  {csn 4576  1_oc1o 8418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-nul 5250

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-nel 3056  df-v 3450  df-dif 3902  df-un 3904  df-nul 4281  df-sn 4577  df-suc 6341  df-1o 8425

This theorem is referenced by: (None)