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Theorem baspartn 20806
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 4206 . . . . . . . . 9 (𝑥𝑃𝑥 ∈ 𝒫 𝑥)
31, 2elind 3831 . . . . . . . 8 (𝑥𝑃𝑥 ∈ (𝑃 ∩ 𝒫 𝑥))
4 elssuni 4499 . . . . . . . 8 (𝑥 ∈ (𝑃 ∩ 𝒫 𝑥) → 𝑥 (𝑃 ∩ 𝒫 𝑥))
53, 4syl 17 . . . . . . 7 (𝑥𝑃𝑥 (𝑃 ∩ 𝒫 𝑥))
6 inidm 3855 . . . . . . . . 9 (𝑥𝑥) = 𝑥
7 ineq2 3841 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥) = (𝑥𝑦))
86, 7syl5eqr 2699 . . . . . . . 8 (𝑥 = 𝑦𝑥 = (𝑥𝑦))
98pweqd 4196 . . . . . . . . . 10 (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 (𝑥𝑦))
109ineq2d 3847 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑃 ∩ 𝒫 𝑥) = (𝑃 ∩ 𝒫 (𝑥𝑦)))
1110unieqd 4478 . . . . . . . 8 (𝑥 = 𝑦 (𝑃 ∩ 𝒫 𝑥) = (𝑃 ∩ 𝒫 (𝑥𝑦)))
128, 11sseq12d 3667 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 (𝑃 ∩ 𝒫 𝑥) ↔ (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
135, 12syl5ibcom 235 . . . . . 6 (𝑥𝑃 → (𝑥 = 𝑦 → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
14 0ss 4005 . . . . . . . 8 ∅ ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))
15 sseq1 3659 . . . . . . . 8 ((𝑥𝑦) = ∅ → ((𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦)) ↔ ∅ ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
1614, 15mpbiri 248 . . . . . . 7 ((𝑥𝑦) = ∅ → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦)))
1716a1i 11 . . . . . 6 (𝑥𝑃 → ((𝑥𝑦) = ∅ → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
1813, 17jaod 394 . . . . 5 (𝑥𝑃 → ((𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
1918ralimdv 2992 . . . 4 (𝑥𝑃 → (∀𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → ∀𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
2019ralimia 2979 . . 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 20799 . . 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 247 1 ((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → 𝑃 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468  TopBasesctb 20797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-uni 4469  df-bases 20798
This theorem is referenced by:  kelac2lem  37951
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