Theorem baspartn 22104

Description: A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
baspartn	⊢ ((𝑃 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → 𝑃 ∈ TopBases)

Distinct variable group:   𝑥,𝑃,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem baspartn

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ 𝑃)
2		pwidg 4555	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ 𝒫 𝑥)
3	1, 2	elind 4128	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ (𝑃 ∩ 𝒫 𝑥))
4		elssuni 4871	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 ∩ 𝒫 𝑥) → 𝑥 ⊆ ∪ (𝑃 ∩ 𝒫 𝑥))
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ⊆ ∪ (𝑃 ∩ 𝒫 𝑥))
6		inidm 4152	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑥) = 𝑥
7		ineq2 4140	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∩ 𝑥) = (𝑥 ∩ 𝑦))
8	6, 7	eqtr3id 2792	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = (𝑥 ∩ 𝑦))
9	8	pweqd 4552	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 (𝑥 ∩ 𝑦))
10	9	ineq2d 4146	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑃 ∩ 𝒫 𝑥) = (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
11	10	unieqd 4853	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ∪ (𝑃 ∩ 𝒫 𝑥) = ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
12	8, 11	sseq12d 3954	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ ∪ (𝑃 ∩ 𝒫 𝑥) ↔ (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
13	5, 12	syl5ibcom 244	. . . . . 6 ⊢ (𝑥 ∈ 𝑃 → (𝑥 = 𝑦 → (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
14		0ss 4330	. . . . . . . 8 ⊢ ∅ ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))
15		sseq1 3946	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ((𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∅ ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
16	14, 15	mpbiri 257	. . . . . . 7 ⊢ ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
17	16	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝑃 → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
18	13, 17	jaod 856	. . . . 5 ⊢ (𝑥 ∈ 𝑃 → ((𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
19	18	ralimdv 3109	. . . 4 ⊢ (𝑥 ∈ 𝑃 → (∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
20	19	ralimia 3085	. . 3 ⊢ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
21	20	adantl 482	. 2 ⊢ ((𝑃 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
22		isbasisg 22097	. . 3 ⊢ (𝑃 ∈ 𝑉 → (𝑃 ∈ TopBases ↔ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
23	22	adantr 481	. 2 ⊢ ((𝑃 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → (𝑃 ∈ TopBases ↔ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
24	21, 23	mpbird 256	1 ⊢ ((𝑃 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → 𝑃 ∈ TopBases)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ∪ cuni 4839  TopBasesctb 22095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-uni 4840  df-bases 22096

This theorem is referenced by:  kelac2lem  40889