Theorem bj-snmoore 34024

Description: A singleton is a Moore collection. (Contributed by BJ, 9-Dec-2021.)

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		snex 5223	. . . 4 ⊢ {𝐴} ∈ V
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
3		unisng 4760	. . . 4 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴)
4		snidg 4504	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
5	3, 4	eqeltrd 2883	. . 3 ⊢ (𝐴 ∈ V → ∪ {𝐴} ∈ {𝐴})
6		df-ne 2985	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
7		sssn 4666	. . . . . . . 8 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
8		biorf 931	. . . . . . . . 9 ⊢ (¬ 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})))
9	8	biimpar 478	. . . . . . . 8 ⊢ ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴})
10	6, 7, 9	syl2anb 597	. . . . . . 7 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴})
11		inteq 4785	. . . . . . . . 9 ⊢ (𝑥 = {𝐴} → ∩ 𝑥 = ∩ {𝐴})
12		intsng 4817	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ∩ {𝐴} = 𝐴)
13		eqtr 2816	. . . . . . . . . 10 ⊢ ((∩ 𝑥 = ∩ {𝐴} ∧ ∩ {𝐴} = 𝐴) → ∩ 𝑥 = 𝐴)
14	13	ex 413	. . . . . . . . 9 ⊢ (∩ 𝑥 = ∩ {𝐴} → (∩ {𝐴} = 𝐴 → ∩ 𝑥 = 𝐴))
15	11, 12, 14	syl2im 40	. . . . . . . 8 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ V → ∩ 𝑥 = 𝐴))
16		intex 5131	. . . . . . . . . 10 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V)
17		elsng 4486	. . . . . . . . . 10 ⊢ (∩ 𝑥 ∈ V → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴))
18	16, 17	sylbi 218	. . . . . . . . 9 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴))
19	18	biimprd 249	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 = 𝐴 → ∩ 𝑥 ∈ {𝐴}))
20	15, 19	sylan9r 509	. . . . . . 7 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 = {𝐴}) → (𝐴 ∈ V → ∩ 𝑥 ∈ {𝐴}))
21	10, 20	syldan 591	. . . . . 6 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → (𝐴 ∈ V → ∩ 𝑥 ∈ {𝐴}))
22	21	ex 413	. . . . 5 ⊢ (𝑥 ≠ ∅ → (𝑥 ⊆ {𝐴} → (𝐴 ∈ V → ∩ 𝑥 ∈ {𝐴})))
23	22	com13 88	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ⊆ {𝐴} → (𝑥 ≠ ∅ → ∩ 𝑥 ∈ {𝐴})))
24	23	imp31 418	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝑥 ⊆ {𝐴}) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ {𝐴})
25	2, 5, 24	bj-ismooredr2 34021	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore)
26		snprc 4560	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
27	26	biimpi 217	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
28		bj-0nmoore 34023	. . . . 5 ⊢ ¬ ∅ ∈ Moore
29	28	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore)
30	27, 29	eqneltrd 2902	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ∈ Moore)
31	30	con4i 114	. 2 ⊢ ({𝐴} ∈ Moore → 𝐴 ∈ V)
32	25, 31	impbii 210	1 ⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∨ wo 842   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  Vcvv 3437   ⊆ wss 3859  ∅c0 4211  {csn 4472  ∪ cuni 4745  ∩ cint 4782  Moorecmoore 34013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-pw 4455  df-sn 4473  df-pr 4475  df-uni 4746  df-int 4783  df-bj-moore 34014

This theorem is referenced by: (None)