Theorem bj-snmoore 37155

Description: A singleton is a Moore collection. See bj-snmooreb 37156 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 4874	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
2		snidg 4610	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3	1, 2	eqeltrd 2831	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} ∈ {𝐴})
4		df-ne 2929	. . . . . 6 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
5		sssn 4775	. . . . . 6 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
6		biorf 936	. . . . . . 7 ⊢ (¬ 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})))
7	6	biimpar 477	. . . . . 6 ⊢ ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴})
8	4, 5, 7	syl2anb 598	. . . . 5 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴})
9		inteq 4898	. . . . . . 7 ⊢ (𝑥 = {𝐴} → ∩ 𝑥 = ∩ {𝐴})
10		intsng 4931	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)
11		eqtr 2751	. . . . . . . 8 ⊢ ((∩ 𝑥 = ∩ {𝐴} ∧ ∩ {𝐴} = 𝐴) → ∩ 𝑥 = 𝐴)
12	11	ex 412	. . . . . . 7 ⊢ (∩ 𝑥 = ∩ {𝐴} → (∩ {𝐴} = 𝐴 → ∩ 𝑥 = 𝐴))
13	9, 10, 12	syl2im 40	. . . . . 6 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑉 → ∩ 𝑥 = 𝐴))
14		intex 5280	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V)
15		elsng 4587	. . . . . . . 8 ⊢ (∩ 𝑥 ∈ V → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴))
16	14, 15	sylbi 217	. . . . . . 7 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴))
17	16	biimprd 248	. . . . . 6 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 = 𝐴 → ∩ 𝑥 ∈ {𝐴}))
18	13, 17	sylan9r 508	. . . . 5 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 = {𝐴}) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}))
19	8, 18	syldan 591	. . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}))
20	19	ancoms 458	. . 3 ⊢ ((𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅) → (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}))
21	20	impcom 407	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ⊆ {𝐴} ∧ 𝑥 ≠ ∅)) → ∩ 𝑥 ∈ {𝐴})
22	3, 21	bj-ismooredr2 37152	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Moore)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436   ⊆ wss 3897  ∅c0 4280  {csn 4573  ∪ cuni 4856  ∩ cint 4895  Moorecmoore 37145

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-pw 4549  df-sn 4574  df-pr 4576  df-uni 4857  df-int 4896  df-bj-moore 37146

This theorem is referenced by:  bj-snmooreb  37156  bj-prmoore  37157