Theorem ontopbas 36389

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 6345	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
2		onelon 6345	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
3	1, 2	anim12dan 619	. . . . . . 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 6351	. . . . . 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 4196	. . . . . . 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 6368	. . . 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 3176	. 2 ⊢ (𝐵 ∈ On → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵)
15		fiinbas 22815	. 2 ⊢ ((𝐵 ∈ On ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
16	14, 15	mpdan 687	1 ⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3044   ∩ cin 3910   ⊆ wss 3911  Oncon0 6320  TopBasesctb 22808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6323  df-on 6324  df-bases 22809

This theorem is referenced by:  onsstopbas  36390  onsuctop  36394