Theorem bj-snmooreb 38305

Description: A singleton is a Moore collection, biconditional version of bj-snmoore 38304. (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 38304	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore)
2		snprc 4680	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 219	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		bj-0nmoore 38303	. . . . 5 ⊢ ¬ ∅ ∈ Moore
5	4	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore)
6	3, 5	eqneltrd 2884	. . 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 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  {csn 4588  Moorecmoore 38294

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-pw 4562  df-sn 4589  df-pr 4591  df-uni 4873  df-int 4913  df-bj-moore 38295

This theorem is used by: (None)