Theorem bj-0int 33055

Description: If 𝐴 is a collection of subsets of 𝑋, like a topology, two equivalent ways to say that arbitrary intersections of elements of 𝐴 relative to 𝑋 belong to some class 𝐵 (in typical applications, 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 3625	. . . . . . . . 9 ⊢ 𝑋 ⊆ V
2		int0 4490	. . . . . . . . 9 ⊢ ∩ ∅ = V
3	1, 2	sseqtr4i 3638	. . . . . . . 8 ⊢ 𝑋 ⊆ ∩ ∅
4		df-ss 3588	. . . . . . . 8 ⊢ (𝑋 ⊆ ∩ ∅ ↔ (𝑋 ∩ ∩ ∅) = 𝑋)
5	3, 4	mpbi 220	. . . . . . 7 ⊢ (𝑋 ∩ ∩ ∅) = 𝑋
6	5	eqcomi 2631	. . . . . 6 ⊢ 𝑋 = (𝑋 ∩ ∩ ∅)
7	6	eleq1i 2692	. . . . 5 ⊢ (𝑋 ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵)
8	7	a1i 11	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝑋 ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
9		eldifsn 4317	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ≠ ∅))
10		sstr2 3610	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝒫 𝑋 → 𝑥 ⊆ 𝒫 𝑋))
11		bj-intss 33053	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝒫 𝑋 → (𝑥 ≠ ∅ → ∩ 𝑥 ⊆ 𝑋))
12	10, 11	syl6 35	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝒫 𝑋 → (𝑥 ≠ ∅ → ∩ 𝑥 ⊆ 𝑋)))
13		elpwi 4168	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴)
14	12, 13	syl11 33	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝑥 ∈ 𝒫 𝐴 → (𝑥 ≠ ∅ → ∩ 𝑥 ⊆ 𝑋)))
15	14	impd 447	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ 𝑋))
16	9, 15	syl5bi 232	. . . . . . 7 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∩ 𝑥 ⊆ 𝑋))
17		df-ss 3588	. . . . . . . . 9 ⊢ (∩ 𝑥 ⊆ 𝑋 ↔ (∩ 𝑥 ∩ 𝑋) = ∩ 𝑥)
18		incom 3805	. . . . . . . . . . 11 ⊢ (∩ 𝑥 ∩ 𝑋) = (𝑋 ∩ ∩ 𝑥)
19	18	eqeq1i 2627	. . . . . . . . . 10 ⊢ ((∩ 𝑥 ∩ 𝑋) = ∩ 𝑥 ↔ (𝑋 ∩ ∩ 𝑥) = ∩ 𝑥)
20		eqcom 2629	. . . . . . . . . 10 ⊢ ((𝑋 ∩ ∩ 𝑥) = ∩ 𝑥 ↔ ∩ 𝑥 = (𝑋 ∩ ∩ 𝑥))
21	19, 20	sylbb 209	. . . . . . . . 9 ⊢ ((∩ 𝑥 ∩ 𝑋) = ∩ 𝑥 → ∩ 𝑥 = (𝑋 ∩ ∩ 𝑥))
22	17, 21	sylbi 207	. . . . . . . 8 ⊢ (∩ 𝑥 ⊆ 𝑋 → ∩ 𝑥 = (𝑋 ∩ ∩ 𝑥))
23		eleq1 2689	. . . . . . . . 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 2965	. . . . 5 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(∩ 𝑥 ∈ 𝐵 ↔ (𝑋 ∩ ∩ 𝑥) ∈ 𝐵))
28		ralbi 3068	. . . . 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 747	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ ((𝑋 ∩ ∩ ∅) ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵)))
31		ancom 466	. . 3 ⊢ (((𝑋 ∩ ∩ ∅) ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵) ↔ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ∧ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
32	30, 31	syl6bb 276	. 2 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ∧ (𝑋 ∩ ∩ ∅) ∈ 𝐵)))
33		0elpw 4834	. . 3 ⊢ ∅ ∈ 𝒫 𝐴
34		inteq 4478	. . . . 5 ⊢ (𝑥 = ∅ → ∩ 𝑥 = ∩ ∅)
35		ineq2 3808	. . . . 5 ⊢ (∩ 𝑥 = ∩ ∅ → (𝑋 ∩ ∩ 𝑥) = (𝑋 ∩ ∩ ∅))
36		eleq1 2689	. . . . 5 ⊢ ((𝑋 ∩ ∩ 𝑥) = (𝑋 ∩ ∩ ∅) → ((𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
37	34, 35, 36	3syl 18	. . . 4 ⊢ (𝑥 = ∅ → ((𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
38	37	bj-raldifsn 33054	. . 3 ⊢ (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ∧ (𝑋 ∩ ∩ ∅) ∈ 𝐵)))
39	33, 38	ax-mp 5	. 2 ⊢ (∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ↔ (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})(𝑋 ∩ ∩ 𝑥) ∈ 𝐵 ∧ (𝑋 ∩ ∩ ∅) ∈ 𝐵))
40	32, 39	syl6bbr 278	1 ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∩ cint 4475

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

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-rab 2921  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-uni 4437  df-int 4476

This theorem is referenced by: (None)