Theorem ist1-2 23469

Description: An alternate characterization of T₁ spaces. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
ist1-2	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝑜,𝐽   𝑜,𝑋,𝑥,𝑦

Proof of Theorem ist1-2

Step	Hyp	Ref	Expression
1		topontop 23035	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		eqid 2769	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	ist1 23443	. . . 4 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽)))
4	3	baib 544	. . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Fre ↔ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽)))
5	1, 4	syl 18	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽)))
6		toponuni 23036	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	raleqdv 3329	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦 ∈ 𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽)))
8	1	adantr 485	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐽 ∈ Top)
9		eltop2 23097	. . . . . 6 ⊢ (𝐽 ∈ Top → ((∪ 𝐽 ∖ {𝑦}) ∈ 𝐽 ↔ ∀𝑥 ∈ (∪ 𝐽 ∖ {𝑦})∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
10	8, 9	syl 18	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ((∪ 𝐽 ∖ {𝑦}) ∈ 𝐽 ↔ ∀𝑥 ∈ (∪ 𝐽 ∖ {𝑦})∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
11	6	eleq2d 2855	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝐽))
12	11	biimpa 481	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ∪ 𝐽)
13	12	snssd 4754	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → {𝑦} ⊆ ∪ 𝐽)
14	2	iscld2 23150	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝑦} ⊆ ∪ 𝐽) → ({𝑦} ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ {𝑦}) ∈ 𝐽))
15	8, 13, 14	syl2anc 595	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ {𝑦}) ∈ 𝐽))
16	6	adantr 485	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
17	16	eleq2d 2855	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽))
18	17	imbi1d 344	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥 ∈ 𝑋 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))) ↔ (𝑥 ∈ ∪ 𝐽 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))))
19		con1b 361	. . . . . . . . 9 ⊢ ((¬ 𝑥 = 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
20		df-ne 2965	. . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
21	20	imbi1i 352	. . . . . . . . 9 ⊢ ((𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (¬ 𝑥 = 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
22		disjsn 4679	. . . . . . . . . . . . . . 15 ⊢ ((𝑜 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑜)
23		elssuni 4905	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
24		reldisj 4416	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ⊆ ∪ 𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
25	23, 24	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ 𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
26	22, 25	bitr3id 288	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∈ 𝐽 → (¬ 𝑦 ∈ 𝑜 ↔ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
27	26	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑜 ∈ 𝐽 → ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
28	27	rexbiia 3116	. . . . . . . . . . . 12 ⊢ (∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
29		rexanali 3125	. . . . . . . . . . . 12 ⊢ (∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
30	28, 29	bitr3i 280	. . . . . . . . . . 11 ⊢ (∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) ↔ ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
31	30	con2bii 360	. . . . . . . . . 10 ⊢ (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
32	31	imbi1i 352	. . . . . . . . 9 ⊢ ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
33	19, 21, 32	3bitr4ri 307	. . . . . . . 8 ⊢ ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
34	33	imbi2i 339	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝑋 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))))
35		eldifsn 4755	. . . . . . . . 9 ⊢ (𝑥 ∈ (∪ 𝐽 ∖ {𝑦}) ↔ (𝑥 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦))
36	35	imbi1i 352	. . . . . . . 8 ⊢ ((𝑥 ∈ (∪ 𝐽 ∖ {𝑦}) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
37		impexp 455	. . . . . . . 8 ⊢ (((𝑥 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (𝑥 ∈ ∪ 𝐽 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))))
38	36, 37	bitri 278	. . . . . . 7 ⊢ ((𝑥 ∈ (∪ 𝐽 ∖ {𝑦}) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (𝑥 ∈ ∪ 𝐽 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))))
39	18, 34, 38	3bitr4g 317	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥 ∈ 𝑋 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ (∪ 𝐽 ∖ {𝑦}) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))))
40	39	ralbidv2 3190	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ (∪ 𝐽 ∖ {𝑦})∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
41	10, 15, 40	3bitr4d 314	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
42	41	ralbidva 3192	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦 ∈ 𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
43		ralcom 3299	. . 3 ⊢ (∀𝑦 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
44	42, 43	bitrdi 290	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦 ∈ 𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
45	5, 7, 44	3bitr2d 310	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  {csn 4591  ∪ cuni 4873  ‘cfv 6533  Topctop 23015  TopOnctopon 23032  Clsdccld 23138  Frect1 23429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6535  df-fv 6541  df-topgen 17492  df-top 23016  df-topon 23033  df-cld 23141  df-t1 23436

This theorem is referenced by:  t1t0  23470  ist1-3  23471  haust1  23474  t1sep2  23491  isr0  23859  tgpt0  24241