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Theorem baspartn 20752
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 4171 . . . . . . . . 9 (𝑥𝑃𝑥 ∈ 𝒫 𝑥)
31, 2elind 3796 . . . . . . . 8 (𝑥𝑃𝑥 ∈ (𝑃 ∩ 𝒫 𝑥))
4 elssuni 4465 . . . . . . . 8 (𝑥 ∈ (𝑃 ∩ 𝒫 𝑥) → 𝑥 (𝑃 ∩ 𝒫 𝑥))
53, 4syl 17 . . . . . . 7 (𝑥𝑃𝑥 (𝑃 ∩ 𝒫 𝑥))
6 inidm 3820 . . . . . . . . 9 (𝑥𝑥) = 𝑥
7 ineq2 3806 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥) = (𝑥𝑦))
86, 7syl5eqr 2669 . . . . . . . 8 (𝑥 = 𝑦𝑥 = (𝑥𝑦))
98pweqd 4161 . . . . . . . . . 10 (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 (𝑥𝑦))
109ineq2d 3812 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑃 ∩ 𝒫 𝑥) = (𝑃 ∩ 𝒫 (𝑥𝑦)))
1110unieqd 4444 . . . . . . . 8 (𝑥 = 𝑦 (𝑃 ∩ 𝒫 𝑥) = (𝑃 ∩ 𝒫 (𝑥𝑦)))
128, 11sseq12d 3632 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 (𝑃 ∩ 𝒫 𝑥) ↔ (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
135, 12syl5ibcom 235 . . . . . 6 (𝑥𝑃 → (𝑥 = 𝑦 → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
14 0ss 3970 . . . . . . . 8 ∅ ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))
15 sseq1 3624 . . . . . . . 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 395 . . . . 5 (𝑥𝑃 → ((𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
1918ralimdv 2962 . . . 4 (𝑥𝑃 → (∀𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅) → ∀𝑦𝑃 (𝑥𝑦) ⊆ (𝑃 ∩ 𝒫 (𝑥𝑦))))
2019ralimia 2949 . . 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 20745 . . 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 247 1 ((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → 𝑃 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1482  wcel 1989  wral 2911  cin 3571  wss 3572  c0 3913  𝒫 cpw 4156   cuni 4434  TopBasesctb 20743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-v 3200  df-dif 3575  df-in 3579  df-ss 3586  df-nul 3914  df-pw 4158  df-uni 4435  df-bases 20744
This theorem is referenced by:  kelac2lem  37460
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