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Theorem ontopbas 32122
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 5717 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
2 onelon 5717 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑦𝐵) → 𝑦 ∈ On)
31, 2anim12dan 881 . . . . . . 7 ((𝐵 ∈ On ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
43ex 450 . . . . . 6 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
5 onin 5723 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥𝑦) ∈ On)
64, 5syl6 35 . . . . 5 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ∈ On))
76anc2ri 580 . . . 4 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ∈ On ∧ 𝐵 ∈ On)))
8 inss1 3817 . . . . . . 7 (𝑥𝑦) ⊆ 𝑥
98jctl 563 . . . . . 6 (𝑥𝐵 → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵))
109adantr 481 . . . . 5 ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵))
1110a1i 11 . . . 4 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → ((𝑥𝑦) ⊆ 𝑥𝑥𝐵)))
12 ontr2 5741 . . . 4 (((𝑥𝑦) ∈ On ∧ 𝐵 ∈ On) → (((𝑥𝑦) ⊆ 𝑥𝑥𝐵) → (𝑥𝑦) ∈ 𝐵))
137, 11, 12syl6c 70 . . 3 (𝐵 ∈ On → ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ∈ 𝐵))
1413ralrimivv 2966 . 2 (𝐵 ∈ On → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵)
15 fiinbas 20696 . 2 ((𝐵 ∈ On ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
1614, 15mpdan 701 1 (𝐵 ∈ On → 𝐵 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2908  cin 3559  wss 3560  Oncon0 5692  TopBasesctb 20689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696  df-bases 20690
This theorem is referenced by:  onsstopbas  32123  onsuctop  32127
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