Theorem bj-snmooreb 38174

Description: A singleton is a Moore collection, biconditional version of bj-snmoore 38173. (Contributed by BJ, 9-Dec-2021.) (Proof shortened by BJ, 10-Apr-2024.)

Assertion

Ref	Expression
bj-snmooreb	⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore)

Proof of Theorem bj-snmooreb

Step	Hyp	Ref	Expression
1		bj-snmoore 38173	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore)
2		snprc 4677	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 219	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		bj-0nmoore 38172	. . . . 5 ⊢ ¬ ∅ ∈ Moore
5	4	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore)
6	3, 5	eqneltrd 2883	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ∈ Moore)
7	6	con4i 115	. 2 ⊢ ({𝐴} ∈ Moore → 𝐴 ∈ V)
8	1, 7	impbii 212	1 ⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1569   ∈ wcel 2146  Vcvv 3453  ∅c0 4280  {csn 4587  Moorecmoore 38163

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-pw 4561  df-sn 4588  df-pr 4590  df-uni 4870  df-int 4910  df-bj-moore 38164

This theorem is used by: (None)