Theorem baspartn 12206

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 19	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ 𝑃)
2		pwidg 3519	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ 𝒫 𝑥)
3	1, 2	elind 3256	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ∈ (𝑃 ∩ 𝒫 𝑥))
4		elssuni 3759	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑃 ∩ 𝒫 𝑥) → 𝑥 ⊆ ∪ (𝑃 ∩ 𝒫 𝑥))
5	3, 4	syl 14	. . . . . . 7 ⊢ (𝑥 ∈ 𝑃 → 𝑥 ⊆ ∪ (𝑃 ∩ 𝒫 𝑥))
6		inidm 3280	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝑥) = 𝑥
7		ineq2 3266	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∩ 𝑥) = (𝑥 ∩ 𝑦))
8	6, 7	syl5eqr 2184	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑥 = (𝑥 ∩ 𝑦))
9	8	pweqd 3510	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 (𝑥 ∩ 𝑦))
10	9	ineq2d 3272	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑃 ∩ 𝒫 𝑥) = (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
11	10	unieqd 3742	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ∪ (𝑃 ∩ 𝒫 𝑥) = ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
12	8, 11	sseq12d 3123	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ ∪ (𝑃 ∩ 𝒫 𝑥) ↔ (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
13	5, 12	syl5ibcom 154	. . . . . 6 ⊢ (𝑥 ∈ 𝑃 → (𝑥 = 𝑦 → (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
14		0ss 3396	. . . . . . . 8 ⊢ ∅ ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))
15		sseq1 3115	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ((𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∅ ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
16	14, 15	mpbiri 167	. . . . . . 7 ⊢ ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
17	16	a1i 9	. . . . . 6 ⊢ (𝑥 ∈ 𝑃 → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
18	13, 17	jaod 706	. . . . 5 ⊢ (𝑥 ∈ 𝑃 → ((𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
19	18	ralimdv 2498	. . . 4 ⊢ (𝑥 ∈ 𝑃 → (∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
20	19	ralimia 2491	. . 3 ⊢ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
21	20	adantl 275	. 2 ⊢ ((𝑃 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦)))
22		isbasisg 12200	. . 3 ⊢ (𝑃 ∈ 𝑉 → (𝑃 ∈ TopBases ↔ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑃 ∩ 𝒫 (𝑥 ∩ 𝑦))))
23	22	adantr 274	. 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 166	1 ⊢ ((𝑃 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → 𝑃 ∈ TopBases)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   = wceq 1331   ∈ wcel 1480  ∀wral 2414   ∩ cin 3065   ⊆ wss 3066  ∅c0 3358  𝒫 cpw 3505  ∪ cuni 3731  TopBasesctb 12198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-uni 3732  df-bases 12199

This theorem is referenced by: (None)