Theorem baspartn 23185

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 23	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ 𝑃)
2		pwidg 4580	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ 𝒫 𝑥)
3	1, 2	elind 4149	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ (𝑃 ∩ 𝒫 𝑥))
4		elssuni 4902	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 ∩ 𝒫 𝑥) → 𝑥 ⊆ ∪ (𝑃 ∩ 𝒫 𝑥))
5	3, 4	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ⊆ ∪ (𝑃 ∩ 𝒫 𝑥))
6		inidm 4175	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑥) = 𝑥
7		ineq2 4163	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∩ 𝑥) = (𝑥 ∩ 𝑦))
8	6, 7	eqtr3id 2811	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = (𝑥 ∩ 𝑦))
9	8	pweqd 4577	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 (𝑥 ∩ 𝑦))
10	9	ineq2d 4169	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑃 ∩ 𝒫 𝑥) = (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
11	10	unieqd 4883	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ∪ (𝑃 ∩ 𝒫 𝑥) = ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
12	8, 11	sseq12d 3967	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ ∪ (𝑃 ∩ 𝒫 𝑥) ↔ (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
13	5, 12	syl5ibcom 248	. . . . . 6 ⊢ (𝑥 ∈ 𝑃 → (𝑥 = 𝑦 → (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
14		0ss 4353	. . . . . . . 8 ⊢ ∅ ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))
15		sseq1 3959	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ((𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∅ ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
16	14, 15	mpbiri 261	. . . . . . 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 873	. . . . 5 ⊢ (𝑥 ∈ 𝑃 → ((𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
19	18	ralimdv 3178	. . . 4 ⊢ (𝑥 ∈ 𝑃 → (∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
20	19	ralimia 3098	. . 3 ⊢ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
21	20	adantl 487	. 2 ⊢ ((𝑃 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
22		isbasisg 23178	. . 3 ⊢ (𝑃 ∈ 𝑉 → (𝑃 ∈ TopBases ↔ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
23	22	adantr 486	. 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 260	1 ⊢ ((𝑃 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → 𝑃 ∈ TopBases)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560  ∪ cuni 4870  TopBasesctb 23176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-in 3909  df-ss 3919  df-nul 4283  df-pw 4562  df-uni 4871  df-bases 23177

This theorem is used by:  kelac2lem  43913