Theorem bj-snmooreb 34977

Description: A singleton is a Moore collection, biconditional version. (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 34976	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore)
2		snprc 4623	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 219	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		bj-0nmoore 34975	. . . . 5 ⊢ ¬ ∅ ∈ Moore
5	4	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore)
6	3, 5	eqneltrd 2853	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ∈ Moore)
7	6	con4i 114	. 2 ⊢ ({𝐴} ∈ Moore → 𝐴 ∈ V)
8	1, 7	impbii 212	1 ⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore)

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1543   ∈ wcel 2110  Vcvv 3401  ∅c0 4227  {csn 4531  Moorecmoore 34966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-pw 4505  df-sn 4532  df-pr 4534  df-uni 4810  df-int 4850  df-bj-moore 34967

This theorem is referenced by: (None)