Theorem bj-snmoore 37812

Description: A singleton is a Moore collection. See bj-snmooreb 37813 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 4892	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
2		snidg 4628	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3	1, 2	eqeltrd 2865	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} ∈ {𝐴})
4		df-ne 2961	. . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
5		sssn 4794	. . . . . 6 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
6		biorf 950	. . . . . . 7 ⊢ (¬ 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})))
7	6	biimpar 483	. . . . . 6 ⊢ ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴})
8	4, 5, 7	syl2anb 610	. . . . 5 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴})
9		inteq 4917	. . . . . . 7 ⊢ (𝑥 = {𝐴} → ∩ 𝑥 = ∩ {𝐴})
10		intsng 4950	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)
11		eqtr 2785	. . . . . . . 8 ⊢ ((∩ 𝑥 = ∩ {𝐴} ∧ ∩ {𝐴} = 𝐴) → ∩ 𝑥 = 𝐴)
12	11	ex 418	. . . . . . 7 ⊢ (∩ 𝑥 = ∩ {𝐴} → (∩ {𝐴} = 𝐴 → ∩ 𝑥 = 𝐴))
13	9, 10, 12	syl2im 41	. . . . . 6 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑉 → ∩ 𝑥 = 𝐴))
14		intex 5316	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V)
15		elsng 4605	. . . . . . . 8 ⊢ (∩ 𝑥 ∈ V → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴))
16	14, 15	sylbi 220	. . . . . . 7 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴))
17	16	biimprd 251	. . . . . 6 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 = 𝐴 → ∩ 𝑥 ∈ {𝐴}))
18	13, 17	sylan9r 518	. . . . 5 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 = {𝐴}) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}))
19	8, 18	syldan 603	. . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}))
20	19	ancoms 464	. . 3 ⊢ ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}))
21	20	impcom 413	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → ∩ 𝑥 ∈ {𝐴})
22	3, 21	bj-ismooredr2 37809	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Moore)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  {csn 4591  ∪ cuni 4874  ∩ cint 4914  Moorecmoore 37802

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-pw 4566  df-sn 4592  df-pr 4594  df-uni 4875  df-int 4915  df-bj-moore 37803

This theorem is used by:  bj-snmooreb  37813  bj-prmoore  37814