Theorem ontopbas 36644

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 6350	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
2		onelon 6350	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
3	1, 2	anim12dan 620	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ On ∧ 𝑦 ∈ On))
4	3	ex 412	. . . . . 6 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ On ∧ 𝑦 ∈ On)))
5		onin 6356	. . . . . 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 556	. . . 4 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∩ 𝑦) ∈ On ∧ 𝐵 ∈ On)))
8		inss1 4191	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
9	8	jctl 523	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝑥 ∩ 𝑦) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵))
10	9	adantr 480	. . . . 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 6373	. . . 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 3179	. 2 ⊢ (𝐵 ∈ On → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵)
15		fiinbas 22908	. 2 ⊢ ((𝐵 ∈ On ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
16	14, 15	mpdan 688	1 ⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3052   ∩ cin 3902   ⊆ wss 3903  Oncon0 6325  TopBasesctb 22901

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329  df-bases 22902

This theorem is referenced by:  onsstopbas  36645  onsuctop  36649