Theorem bj-1nel0 36955

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 8526	. . . 4 ⊢ 1o ≠ ∅
2	1	neii 2942	. . 3 ⊢ ¬ 1o = ∅
3		elsni 4643	. . 3 ⊢ (1o ∈ {∅} → 1o = ∅)
4	2, 3	mto 197	. 2 ⊢ ¬ 1o ∈ {∅}
5	4	nelir 3049	1 ⊢ 1o ∉ {∅}

Syntax hints:   = wceq 1540   ∈ wcel 2108   ∉ wnel 3046  ∅c0 4333  {csn 4626  1_oc1o 8499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-suc 6390  df-1o 8506

This theorem is referenced by: (None)