Theorem bj-snmoore 33068

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 4908	. . . 4 ⊢ {𝐴} ∈ V
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
3		unisng 4452	. . . 4 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴)
4		snidg 4206	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴})
5	3, 4	eqeltrd 2701	. . 3 ⊢ (𝐴 ∈ V → ∪ {𝐴} ∈ {𝐴})
6		df-ne 2795	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅)
7		sssn 4358	. . . . . . . 8 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
8		biorf 420	. . . . . . . . 9 ⊢ (¬ 𝑥 = ∅ → (𝑥 = {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})))
9	8	biimpar 502	. . . . . . . 8 ⊢ ((¬ 𝑥 = ∅ ∧ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) → 𝑥 = {𝐴})
10	6, 7, 9	syl2anb 496	. . . . . . 7 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → 𝑥 = {𝐴})
11		inteq 4478	. . . . . . . . 9 ⊢ (𝑥 = {𝐴} → ∩ 𝑥 = ∩ {𝐴})
12		intsng 4512	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ∩ {𝐴} = 𝐴)
13		eqtr 2641	. . . . . . . . . 10 ⊢ ((∩ 𝑥 = ∩ {𝐴} ∧ ∩ {𝐴} = 𝐴) → ∩ 𝑥 = 𝐴)
14	13	ex 450	. . . . . . . . 9 ⊢ (∩ 𝑥 = ∩ {𝐴} → (∩ {𝐴} = 𝐴 → ∩ 𝑥 = 𝐴))
15	11, 12, 14	syl2im 40	. . . . . . . 8 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ V → ∩ 𝑥 = 𝐴))
16		intex 4820	. . . . . . . . . 10 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V)
17		elsng 4191	. . . . . . . . . 10 ⊢ (∩ 𝑥 ∈ V → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴))
18	16, 17	sylbi 207	. . . . . . . . 9 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 ∈ {𝐴} ↔ ∩ 𝑥 = 𝐴))
19	18	biimprd 238	. . . . . . . 8 ⊢ (𝑥 ≠ ∅ → (∩ 𝑥 = 𝐴 → ∩ 𝑥 ∈ {𝐴}))
20	15, 19	sylan9r 690	. . . . . . 7 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 = {𝐴}) → (𝐴 ∈ V → ∩ 𝑥 ∈ {𝐴}))
21	10, 20	syldan 487	. . . . . 6 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ {𝐴}) → (𝐴 ∈ V → ∩ 𝑥 ∈ {𝐴}))
22	21	ex 450	. . . . 5 ⊢ (𝑥 ≠ ∅ → (𝑥 ⊆ {𝐴} → (𝐴 ∈ V → ∩ 𝑥 ∈ {𝐴})))
23	22	com13 88	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ⊆ {𝐴} → (𝑥 ≠ ∅ → ∩ 𝑥 ∈ {𝐴})))
24	23	imp31 448	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝑥 ⊆ {𝐴}) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ {𝐴})
25	2, 5, 24	bj-ismooredr2 33065	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore)
26		snprc 4253	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
27	26	biimpi 206	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
28		bj-0nmoore 33067	. . . . 5 ⊢ ¬ ∅ ∈ Moore
29	28	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore)
30	27, 29	eqneltrd 2720	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ∈ Moore)
31	30	con4i 113	. 2 ⊢ ({𝐴} ∈ Moore → 𝐴 ∈ V)
32	25, 31	impbii 199	1 ⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  {csn 4177  ∪ cuni 4436  ∩ cint 4475  Moorecmoore 33057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-int 4476  df-bj-moore 33058

This theorem is referenced by: (None)