Theorem bj-snmooreb 37258

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 37257	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore)
2		snprc 4672	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 216	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		bj-0nmoore 37256	. . . . 5 ⊢ ¬ ∅ ∈ Moore
5	4	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore)
6	3, 5	eqneltrd 2854	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ∈ Moore)
7	6	con4i 114	. 2 ⊢ ({𝐴} ∈ Moore → 𝐴 ∈ V)
8	1, 7	impbii 209	1 ⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ∅c0 4283  {csn 4578  Moorecmoore 37247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-pw 4554  df-sn 4579  df-pr 4581  df-uni 4862  df-int 4901  df-bj-moore 37248

This theorem is referenced by: (None)