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Theorem baspartn 22962
Description: A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
baspartn ((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → 𝑃 ∈ TopBases)
Distinct variable group:   𝑥,𝑃,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem baspartn
StepHypRef Expression
1 id 22 . . . . . . . . 9 (𝑥𝑃𝑥𝑃)
2 pwidg 4619 . . . . . . . . 9 (𝑥𝑃𝑥 ∈ 𝒫 𝑥)
31, 2elind 4199 . . . . . . . 8 (𝑥𝑃𝑥 ∈ (𝑃 ∩ 𝒫 𝑥))
4 elssuni 4936 . . . . . . . 8 (𝑥 ∈ (𝑃 ∩ 𝒫 𝑥) → 𝑥 (𝑃 ∩ 𝒫 𝑥))
53, 4syl 17 . . . . . . 7 (𝑥𝑃𝑥 (𝑃 ∩ 𝒫 𝑥))
6 inidm 4226 . . . . . . . . 9 (𝑥𝑥) = 𝑥
7 ineq2 4213 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥) = (𝑥𝑦))
86, 7eqtr3id 2790 . . . . . . . 8 (𝑥 = 𝑦𝑥 = (𝑥𝑦))
98pweqd 4616 . . . . . . . . . 10 (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 (𝑥𝑦))
109ineq2d 4219 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑃 ∩ 𝒫 𝑥) = (𝑃 ∩ 𝒫 (𝑥𝑦)))
1110unieqd 4919 . . . . . . . 8 (𝑥 = 𝑦 (𝑃 ∩ 𝒫 𝑥) = (𝑃 ∩ 𝒫 (𝑥𝑦)))
128, 11sseq12d 4016 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 (𝑃 ∩ 𝒫 𝑥) ↔ (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
135, 12syl5ibcom 245 . . . . . 6 (𝑥𝑃 → (𝑥 = 𝑦 → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
14 0ss 4399 . . . . . . . 8 ∅ ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))
15 sseq1 4008 . . . . . . . 8 ((𝑥𝑦) = ∅ → ((𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦)) ↔ ∅ ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
1614, 15mpbiri 258 . . . . . . 7 ((𝑥𝑦) = ∅ → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦)))
1716a1i 11 . . . . . 6 (𝑥𝑃 → ((𝑥𝑦) = ∅ → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
1813, 17jaod 859 . . . . 5 (𝑥𝑃 → ((𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
1918ralimdv 3168 . . . 4 (𝑥𝑃 → (∀𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → ∀𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
2019ralimia 3079 . . 3 (∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → ∀𝑥𝑃𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦)))
2120adantl 481 . 2 ((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → ∀𝑥𝑃𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦)))
22 isbasisg 22955 . . 3 (𝑃𝑉 → (𝑃 ∈ TopBases ↔ ∀𝑥𝑃𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
2322adantr 480 . 2 ((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → (𝑃 ∈ TopBases ↔ ∀𝑥𝑃𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
2421, 23mpbird 257 1 ((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → 𝑃 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1539  wcel 2107  wral 3060  cin 3949  wss 3950  c0 4332  𝒫 cpw 4599   cuni 4906  TopBasesctb 22953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-in 3957  df-ss 3967  df-nul 4333  df-pw 4601  df-uni 4907  df-bases 22954
This theorem is referenced by:  kelac2lem  43081
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