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Theorem ontopbas 33794
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 6197 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
2 onelon 6197 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦𝐵) → 𝑦 ∈ On)
31, 2anim12dan 621 . . . . . . 7 ((𝐵 ∈ On ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
43ex 416 . . . . . 6 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
5 onin 6203 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦) ∈ On)
64, 5syl6 35 . . . . 5 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ∈ On))
76anc2ri 560 . . . 4 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∈ On ∧ 𝐵 ∈ On)))
8 inss1 4188 . . . . . . 7 (𝑥𝑦) ⊆ 𝑥
98jctl 527 . . . . . 6 (𝑥𝐵 → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵))
109adantr 484 . . . . 5 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵))
1110a1i 11 . . . 4 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵)))
12 ontr2 6219 . . . 4 (((𝑥𝑦) ∈ On ∧ 𝐵 ∈ On) → (((𝑥𝑦) ⊆ 𝑥𝑥𝐵) → (𝑥𝑦) ∈ 𝐵))
137, 11, 12syl6c 70 . . 3 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ∈ 𝐵))
1413ralrimivv 3184 . 2 (𝐵 ∈ On → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵)
15 fiinbas 21546 . 2 ((𝐵 ∈ On ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
1614, 15mpdan 686 1 (𝐵 ∈ On → 𝐵 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wral 3132  cin 3917  wss 3918  Oncon0 6172  TopBasesctb 21539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-tr 5154  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-ord 6175  df-on 6176  df-bases 21540
This theorem is referenced by:  onsstopbas  33795  onsuctop  33799
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