Theorem bj-snmoore 36297

Description: A singleton is a Moore collection. See bj-snmooreb 36298 for a biconditional version. (Contributed by BJ, 10-Apr-2024.)

Assertion

Ref	Expression
bj-snmoore	⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Moore)

Proof of Theorem bj-snmoore

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unisng 4928	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
2		snidg 4661	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3	1, 2	eqeltrd 2831	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} ∈ {𝐴})
4		df-ne 2939	. . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
5		sssn 4828	. . . . . 6 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
6		biorf 933	. . . . . . 7 ⊢ (¬ 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})))
7	6	biimpar 476	. . . . . 6 ⊢ ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴})
8	4, 5, 7	syl2anb 596	. . . . 5 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴})
9		inteq 4952	. . . . . . 7 ⊢ (𝑥 = {𝐴} → ∩ 𝑥 = ∩ {𝐴})
10		intsng 4988	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)
11		eqtr 2753	. . . . . . . 8 ⊢ ((∩ 𝑥 = ∩ {𝐴} ∧ ∩ {𝐴} = 𝐴) → ∩ 𝑥 = 𝐴)
12	11	ex 411	. . . . . . 7 ⊢ (∩ 𝑥 = ∩ {𝐴} → (∩ {𝐴} = 𝐴 → ∩ 𝑥 = 𝐴))
13	9, 10, 12	syl2im 40	. . . . . 6 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑉 → ∩ 𝑥 = 𝐴))
14		intex 5336	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V)
15		elsng 4641	. . . . . . . 8 ⊢ (∩ 𝑥 ∈ V → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴))
16	14, 15	sylbi 216	. . . . . . 7 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴))
17	16	biimprd 247	. . . . . 6 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 = 𝐴 → ∩ 𝑥 ∈ {𝐴}))
18	13, 17	sylan9r 507	. . . . 5 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 = {𝐴}) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}))
19	8, 18	syldan 589	. . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}))
20	19	ancoms 457	. . 3 ⊢ ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}))
21	20	impcom 406	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → ∩ 𝑥 ∈ {𝐴})
22	3, 21	bj-ismooredr2 36294	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Moore)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  Vcvv 3472   ⊆ wss 3947  ∅c0 4321  {csn 4627  ∪ cuni 4907  ∩ cint 4949  Moorecmoore 36287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-pw 4603  df-sn 4628  df-pr 4630  df-uni 4908  df-int 4950  df-bj-moore 36288

This theorem is referenced by:  bj-snmooreb  36298  bj-prmoore  36299