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Theorem ontopbas 33673
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 6209 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
2 onelon 6209 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦𝐵) → 𝑦 ∈ On)
31, 2anim12dan 618 . . . . . . 7 ((𝐵 ∈ On ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
43ex 413 . . . . . 6 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
5 onin 6215 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦) ∈ On)
64, 5syl6 35 . . . . 5 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ∈ On))
76anc2ri 557 . . . 4 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∈ On ∧ 𝐵 ∈ On)))
8 inss1 4202 . . . . . . 7 (𝑥𝑦) ⊆ 𝑥
98jctl 524 . . . . . 6 (𝑥𝐵 → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵))
109adantr 481 . . . . 5 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵))
1110a1i 11 . . . 4 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵)))
12 ontr2 6231 . . . 4 (((𝑥𝑦) ∈ On ∧ 𝐵 ∈ On) → (((𝑥𝑦) ⊆ 𝑥𝑥𝐵) → (𝑥𝑦) ∈ 𝐵))
137, 11, 12syl6c 70 . . 3 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ∈ 𝐵))
1413ralrimivv 3187 . 2 (𝐵 ∈ On → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵)
15 fiinbas 21488 . 2 ((𝐵 ∈ On ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
1614, 15mpdan 683 1 (𝐵 ∈ On → 𝐵 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wral 3135  cin 3932  wss 3933  Oncon0 6184  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  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-bases 21482
This theorem is referenced by:  onsstopbas  33674  onsuctop  33678
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