Theorem ontopbas 33889

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.

Step	Hyp	Ref	Expression
1		onelon 6184	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
2		onelon 6184	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
3	1, 2	anim12dan 621	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
4	3	ex 416	. . . . . 6 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
5		onin 6190	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∩ 𝑦) ∈ On)
6	4, 5	syl6 35	. . . . 5 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∩ 𝑦) ∈ On))
7	6	anc2ri 560	. . . 4 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∩ 𝑦) ∈ On ∧ 𝐵 ∈ On)))
8		inss1 4155	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
9	8	jctl 527	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝑥 ∩ 𝑦) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵))
10	9	adantr 484	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∩ 𝑦) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵))
11	10	a1i 11	. . . 4 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∩ 𝑦) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵)))
12		ontr2 6206	. . . 4 ⊢ (((𝑥 ∩ 𝑦) ∈ On ∧ 𝐵 ∈ On) → (((𝑥 ∩ 𝑦) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∩ 𝑦) ∈ 𝐵))
13	7, 11, 12	syl6c 70	. . 3 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∩ 𝑦) ∈ 𝐵))
14	13	ralrimivv 3155	. 2 ⊢ (𝐵 ∈ On → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵)
15		fiinbas 21557	. 2 ⊢ ((𝐵 ∈ On ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
16	14, 15	mpdan 686	1 ⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  ∀wral 3106   ∩ cin 3880   ⊆ wss 3881  Oncon0 6159  TopBasesctb 21550

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-bases 21551

This theorem is referenced by:  onsstopbas  33890  onsuctop  33894