Theorem bj-1nel0 32941

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 7575	. . . 4 ⊢ 1𝑜 ≠ ∅
2	1	neii 2796	. . 3 ⊢ ¬ 1𝑜 = ∅
3		elsni 4194	. . 3 ⊢ (1𝑜 ∈ {∅} → 1𝑜 = ∅)
4	2, 3	mto 188	. 2 ⊢ ¬ 1𝑜 ∈ {∅}
5	4	nelir 2900	1 ⊢ 1𝑜 ∉ {∅}

Syntax hints:   = wceq 1483   ∈ wcel 1990   ∉ wnel 2897  ∅c0 3915  {csn 4177  1_𝑜c1o 7553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-suc 5729  df-1o 7560

This theorem is referenced by: (None)