Theorem bj-1nel0 37157

Description: 1_o does not belong to {∅}. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-1nel0	⊢ 1o ∉ {∅}

Proof of Theorem bj-1nel0

Step	Hyp	Ref	Expression
1		1n0 8417	. . . 4 ⊢ 1o ≠ ∅
2	1	neii 2935	. . 3 ⊢ ¬ 1o = ∅
3		elsni 4598	. . 3 ⊢ (1o ∈ {∅} → 1o = ∅)
4	2, 3	mto 197	. 2 ⊢ ¬ 1o ∈ {∅}
5	4	nelir 3040	1 ⊢ 1o ∉ {∅}

Syntax hints:   = wceq 1542   ∈ wcel 2114   ∉ wnel 3037  ∅c0 4286  {csn 4581  1_oc1o 8392

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5252

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-nel 3038  df-v 3443  df-dif 3905  df-un 3907  df-nul 4287  df-sn 4582  df-suc 6324  df-1o 8399

This theorem is referenced by: (None)