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Theorem isbasis3g 20801
Description: Express the predicate "𝐵 is a basis for a topology." Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
isbasis3g (𝐵𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))
Distinct variable group:   𝑥,𝑤,𝑦,𝑧,𝐵
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem isbasis3g
StepHypRef Expression
1 isbasis2g 20800 . 2 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
2 elssuni 4499 . . . . . 6 (𝑥𝐵𝑥 𝐵)
32rgen 2951 . . . . 5 𝑥𝐵 𝑥 𝐵
4 eluni2 4472 . . . . . . 7 (𝑥 𝐵 ↔ ∃𝑦𝐵 𝑥𝑦)
54biimpi 206 . . . . . 6 (𝑥 𝐵 → ∃𝑦𝐵 𝑥𝑦)
65rgen 2951 . . . . 5 𝑥 𝐵𝑦𝐵 𝑥𝑦
73, 6pm3.2i 470 . . . 4 (∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦)
87biantrur 526 . . 3 (∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ ((∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦) ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
9 df-3an 1056 . . 3 ((∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))) ↔ ((∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦) ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
108, 9bitr4i 267 . 2 (∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ (∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
111, 10syl6bb 276 1 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wcel 2030  wral 2941  wrex 2942  cin 3606  wss 3607   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-3an 1056  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-in 3614  df-ss 3621  df-pw 4193  df-uni 4469  df-bases 20798
This theorem is referenced by: (None)
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