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Theorem ontopbas 32654
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 5861 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
2 onelon 5861 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦𝐵) → 𝑦 ∈ On)
31, 2anim12dan 918 . . . . . . 7 ((𝐵 ∈ On ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
43ex 449 . . . . . 6 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
5 onin 5867 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦) ∈ On)
64, 5syl6 35 . . . . 5 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ∈ On))
76anc2ri 582 . . . 4 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∈ On ∧ 𝐵 ∈ On)))
8 inss1 3941 . . . . . . 7 (𝑥𝑦) ⊆ 𝑥
98jctl 565 . . . . . 6 (𝑥𝐵 → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵))
109adantr 472 . . . . 5 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵))
1110a1i 11 . . . 4 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵)))
12 ontr2 5885 . . . 4 (((𝑥𝑦) ∈ On ∧ 𝐵 ∈ On) → (((𝑥𝑦) ⊆ 𝑥𝑥𝐵) → (𝑥𝑦) ∈ 𝐵))
137, 11, 12syl6c 70 . . 3 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ∈ 𝐵))
1413ralrimivv 3072 . 2 (𝐵 ∈ On → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵)
15 fiinbas 20879 . 2 ((𝐵 ∈ On ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
1614, 15mpdan 705 1 (𝐵 ∈ On → 𝐵 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2103  wral 3014  cin 3679  wss 3680  Oncon0 5836  TopBasesctb 20872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-tr 4861  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-ord 5839  df-on 5840  df-bases 20873
This theorem is referenced by:  onsstopbas  32655  onsuctop  32659
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