Theorem bj-1nel0 33066

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 7620	. . . 4 ⊢ 1𝑜 ≠ ∅
2	1	neii 2825	. . 3 ⊢ ¬ 1𝑜 = ∅
3		elsni 4227	. . 3 ⊢ (1𝑜 ∈ {∅} → 1𝑜 = ∅)
4	2, 3	mto 188	. 2 ⊢ ¬ 1𝑜 ∈ {∅}
5	4	nelir 2929	1 ⊢ 1𝑜 ∉ {∅}

Syntax hints:   = wceq 1523   ∈ wcel 2030   ∉ wnel 2926  ∅c0 3948  {csn 4210  1_𝑜c1o 7598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-suc 5767  df-1o 7605

This theorem is referenced by: (None)