Theorem bj-snmooreb 37445

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 37444	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore)
2		snprc 4662	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 216	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		bj-0nmoore 37443	. . . . 5 ⊢ ¬ ∅ ∈ Moore
5	4	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore)
6	3, 5	eqneltrd 2857	. . 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 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  {csn 4568  Moorecmoore 37434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-pw 4544  df-sn 4569  df-pr 4571  df-uni 4852  df-int 4891  df-bj-moore 37435

This theorem is referenced by: (None)