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Theorem ontopbas 36430
Description: An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
Assertion
Ref Expression
ontopbas (𝐵 ∈ On → 𝐵 ∈ TopBases)

Proof of Theorem ontopbas
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onelon 6408 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
2 onelon 6408 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦𝐵) → 𝑦 ∈ On)
31, 2anim12dan 619 . . . . . . 7 ((𝐵 ∈ On ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
43ex 412 . . . . . 6 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
5 onin 6414 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦) ∈ On)
64, 5syl6 35 . . . . 5 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ∈ On))
76anc2ri 556 . . . 4 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∈ On ∧ 𝐵 ∈ On)))
8 inss1 4236 . . . . . . 7 (𝑥𝑦) ⊆ 𝑥
98jctl 523 . . . . . 6 (𝑥𝐵 → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵))
109adantr 480 . . . . 5 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵))
1110a1i 11 . . . 4 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵)))
12 ontr2 6430 . . . 4 (((𝑥𝑦) ∈ On ∧ 𝐵 ∈ On) → (((𝑥𝑦) ⊆ 𝑥𝑥𝐵) → (𝑥𝑦) ∈ 𝐵))
137, 11, 12syl6c 70 . . 3 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ∈ 𝐵))
1413ralrimivv 3199 . 2 (𝐵 ∈ On → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵)
15 fiinbas 22960 . 2 ((𝐵 ∈ On ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
1614, 15mpdan 687 1 (𝐵 ∈ On → 𝐵 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2107  wral 3060  cin 3949  wss 3950  Oncon0 6383  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  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-bases 22954
This theorem is referenced by:  onsstopbas  36431  onsuctop  36435
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