Theorem bj-0nel1 33461

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 7847	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3056	. . 3 ⊢ ¬ ∅ = 1o
3		0ex 5016	. . . 4 ⊢ ∅ ∈ V
4	3	elsn 4414	. . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o)
5	2, 4	mtbir 315	. 2 ⊢ ¬ ∅ ∈ {1o}
6	5	nelir 3105	1 ⊢ ∅ ∉ {1o}

Syntax hints:   = wceq 1656   ∈ wcel 2164   ∉ wnel 3102  ∅c0 4146  {csn 4399  1_oc1o 7824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-v 3416  df-dif 3801  df-un 3803  df-nul 4147  df-sn 4400  df-suc 5973  df-1o 7831

This theorem is referenced by: (None)