Theorem bj-0int 37124

Description: If 𝐴 is a collection of subsets of 𝑋, like a Moore collection or a topology, two equivalent ways to say that arbitrary intersections of elements of 𝐴 relative to 𝑋 belong to some class 𝐵: the LHS singles out the empty intersection (the empty intersection relative to 𝑋 is 𝑋 and the intersection of a nonempty family of subsets of 𝑋 is included in 𝑋, so there is no need to intersect it with 𝑋). In typical applications, 𝐵 is 𝐴 itself. (Contributed by BJ, 7-Dec-2021.)

Assertion

Ref	Expression
bj-0int	⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑋

Proof of Theorem bj-0int

Step	Hyp	Ref	Expression
1		ssv 3988	. . . . . . . . 9 ⊢ 𝑋 ⊆ V
2		int0 4943	. . . . . . . . 9 ⊢ ∩ ∅ = V
3	1, 2	sseqtrri 4013	. . . . . . . 8 ⊢ 𝑋 ⊆ ∩ ∅
4		dfss2 3949	. . . . . . . 8 ⊢ (𝑋 ⊆ ∩ ∅ ↔ (𝑋 ∩ ∩ ∅) = 𝑋)
5	3, 4	mpbi 230	. . . . . . 7 ⊢ (𝑋 ∩ ∩ ∅) = 𝑋
6	5	eqcomi 2745	. . . . . 6 ⊢ 𝑋 = (𝑋 ∩ ∩ ∅)
7	6	eleq1i 2826	. . . . 5 ⊢ (𝑋 ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵)
8	7	a1i 11	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝑋 ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
9		eldifsn 4767	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ≠ ∅))
10		sstr2 3970	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝒫 𝑋 → 𝑥 ⊆ 𝒫 𝑋))
11		intss2 5089	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝒫 𝑋 → (𝑥 ≠ ∅ → ∩ 𝑥 ⊆ 𝑋))
12	10, 11	syl6 35	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝒫 𝑋 → (𝑥 ≠ ∅ → ∩ 𝑥 ⊆ 𝑋)))
13		elpwi 4587	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
14	12, 13	syl11 33	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝑥 ∈ 𝒫 𝐴 → (𝑥 ≠ ∅ → ∩ 𝑥 ⊆ 𝑋)))
15	14	impd 410	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ 𝑋))
16	9, 15	biimtrid 242	. . . . . . 7 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∩ 𝑥 ⊆ 𝑋))
17		dfss2 3949	. . . . . . . . 9 ⊢ (∩ 𝑥 ⊆ 𝑋 ↔ (∩ 𝑥 ∩ 𝑋) = ∩ 𝑥)
18		incom 4189	. . . . . . . . . . 11 ⊢ (∩ 𝑥 ∩ 𝑋) = (𝑋 ∩ ∩ 𝑥)
19	18	eqeq1i 2741	. . . . . . . . . 10 ⊢ ((∩ 𝑥 ∩ 𝑋) = ∩ 𝑥 ↔ (𝑋 ∩ ∩ 𝑥) = ∩ 𝑥)
20		eqcom 2743	. . . . . . . . . 10 ⊢ ((𝑋 ∩ ∩ 𝑥) = ∩ 𝑥 ↔ ∩ 𝑥 = (𝑋 ∩ ∩ 𝑥))
21	19, 20	sylbb 219	. . . . . . . . 9 ⊢ ((∩ 𝑥 ∩ 𝑋) = ∩ 𝑥 → ∩ 𝑥 = (𝑋 ∩ ∩ 𝑥))
22	17, 21	sylbi 217	. . . . . . . 8 ⊢ (∩ 𝑥 ⊆ 𝑋 → ∩ 𝑥 = (𝑋 ∩ ∩ 𝑥))
23		eleq1 2823	. . . . . . . . 9 ⊢ (∩ 𝑥 = (𝑋 ∩ ∩ 𝑥) → (∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵))
24	23	a1i 11	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (∩ 𝑥 = (𝑋 ∩ ∩ 𝑥) → (∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵)))
25	22, 24	syl5 34	. . . . . . 7 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (∩ 𝑥 ⊆ 𝑋 → (∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵)))
26	16, 25	syld 47	. . . . . 6 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → (∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵)))
27	26	ralrimiv 3132	. . . . 5 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵))
28		ralbi 3093	. . . . 5 ⊢ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵) → (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵))
29	27, 28	syl 17	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵))
30	8, 29	anbi12d 632	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ ((𝑋 ∩ ∩ ∅) ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵)))
31	30	biancomd 463	. 2 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ∧ (𝑋 ∩ ∩ ∅) ∈ 𝐵)))
32		0elpw 5331	. . 3 ⊢ ∅ ∈ 𝒫 𝐴
33		inteq 4930	. . . . 5 ⊢ (𝑥 = ∅ → ∩ 𝑥 = ∩ ∅)
34		ineq2 4194	. . . . 5 ⊢ (∩ 𝑥 = ∩ ∅ → (𝑋 ∩ ∩ 𝑥) = (𝑋 ∩ ∩ ∅))
35		eleq1 2823	. . . . 5 ⊢ ((𝑋 ∩ ∩ 𝑥) = (𝑋 ∩ ∩ ∅) → ((𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
36	33, 34, 35	3syl 18	. . . 4 ⊢ (𝑥 = ∅ → ((𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
37	36	bj-raldifsn 37123	. . 3 ⊢ (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ∧ (𝑋 ∩ ∩ ∅) ∈ 𝐵)))
38	32, 37	ax-mp 5	. 2 ⊢ (∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ∧ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
39	31, 38	bitr4di 289	1 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  Vcvv 3464   ∖ cdif 3928   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  {csn 4606  ∩ cint 4927

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-pw 4582  df-sn 4607  df-uni 4889  df-int 4928

This theorem is referenced by: (None)