Theorem bj-1nel0 36920

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 8544	. . . 4 ⊢ 1o ≠ ∅
2	1	neii 2948	. . 3 ⊢ ¬ 1o = ∅
3		elsni 4665	. . 3 ⊢ (1o ∈ {∅} → 1o = ∅)
4	2, 3	mto 197	. 2 ⊢ ¬ 1o ∈ {∅}
5	4	nelir 3055	1 ⊢ 1o ∉ {∅}

Syntax hints:   = wceq 1537   ∈ wcel 2108   ∉ wnel 3052  ∅c0 4352  {csn 4648  1_oc1o 8515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-nel 3053  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-sn 4649  df-suc 6401  df-1o 8522

This theorem is referenced by: (None)