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Theorem baspartn 21490
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 4552 . . . . . . . . 9 (𝑥𝑃𝑥 ∈ 𝒫 𝑥)
31, 2elind 4168 . . . . . . . 8 (𝑥𝑃𝑥 ∈ (𝑃 ∩ 𝒫 𝑥))
4 elssuni 4859 . . . . . . . 8 (𝑥 ∈ (𝑃 ∩ 𝒫 𝑥) → 𝑥 (𝑃 ∩ 𝒫 𝑥))
53, 4syl 17 . . . . . . 7 (𝑥𝑃𝑥 (𝑃 ∩ 𝒫 𝑥))
6 inidm 4192 . . . . . . . . 9 (𝑥𝑥) = 𝑥
7 ineq2 4180 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥) = (𝑥𝑦))
86, 7syl5eqr 2867 . . . . . . . 8 (𝑥 = 𝑦𝑥 = (𝑥𝑦))
98pweqd 4540 . . . . . . . . . 10 (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 (𝑥𝑦))
109ineq2d 4186 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑃 ∩ 𝒫 𝑥) = (𝑃 ∩ 𝒫 (𝑥𝑦)))
1110unieqd 4840 . . . . . . . 8 (𝑥 = 𝑦 (𝑃 ∩ 𝒫 𝑥) = (𝑃 ∩ 𝒫 (𝑥𝑦)))
128, 11sseq12d 3997 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 (𝑃 ∩ 𝒫 𝑥) ↔ (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
135, 12syl5ibcom 246 . . . . . 6 (𝑥𝑃 → (𝑥 = 𝑦 → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
14 0ss 4347 . . . . . . . 8 ∅ ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))
15 sseq1 3989 . . . . . . . 8 ((𝑥𝑦) = ∅ → ((𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦)) ↔ ∅ ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
1614, 15mpbiri 259 . . . . . . 7 ((𝑥𝑦) = ∅ → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦)))
1716a1i 11 . . . . . 6 (𝑥𝑃 → ((𝑥𝑦) = ∅ → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
1813, 17jaod 853 . . . . 5 (𝑥𝑃 → ((𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
1918ralimdv 3175 . . . 4 (𝑥𝑃 → (∀𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → ∀𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
2019ralimia 3155 . . 3 (∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → ∀𝑥𝑃𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦)))
2120adantl 482 . 2 ((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → ∀𝑥𝑃𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦)))
22 isbasisg 21483 . . 3 (𝑃𝑉 → (𝑃 ∈ TopBases ↔ ∀𝑥𝑃𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
2322adantr 481 . 2 ((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → (𝑃 ∈ TopBases ↔ ∀𝑥𝑃𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
2421, 23mpbird 258 1 ((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → 𝑃 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3135  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830  TopBasesctb 21481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-pw 4537  df-uni 4831  df-bases 21482
This theorem is referenced by:  kelac2lem  39542
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