Theorem ist1-2 21151

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 20718	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		eqid 2622	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	ist1 21125	. . . 4 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽)))
4	3	baib 944	. . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Fre ↔ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽)))
5	1, 4	syl 17	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽)))
6		toponuni 20719	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	raleqdv 3144	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦 ∈ 𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽)))
8	1	adantr 481	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐽 ∈ Top)
9		eltop2 20779	. . . . . 6 ⊢ (𝐽 ∈ Top → ((∪ 𝐽 ∖ {𝑦}) ∈ 𝐽 ↔ ∀𝑥 ∈ (∪ 𝐽 ∖ {𝑦})∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
10	8, 9	syl 17	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ((∪ 𝐽 ∖ {𝑦}) ∈ 𝐽 ↔ ∀𝑥 ∈ (∪ 𝐽 ∖ {𝑦})∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
11	6	eleq2d 2687	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝐽))
12	11	biimpa 501	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ∪ 𝐽)
13	12	snssd 4340	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → {𝑦} ⊆ ∪ 𝐽)
14	2	iscld2 20832	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝑦} ⊆ ∪ 𝐽) → ({𝑦} ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ {𝑦}) ∈ 𝐽))
15	8, 13, 14	syl2anc 693	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ {𝑦}) ∈ 𝐽))
16	6	adantr 481	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
17	16	eleq2d 2687	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽))
18	17	imbi1d 331	. . . . . . 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 348	. . . . . . . . 9 ⊢ ((¬ 𝑥 = 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
20		df-ne 2795	. . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
21	20	imbi1i 339	. . . . . . . . 9 ⊢ ((𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (¬ 𝑥 = 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
22		disjsn 4246	. . . . . . . . . . . . . . 15 ⊢ ((𝑜 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑜)
23		elssuni 4467	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
24		reldisj 4020	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ⊆ ∪ 𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
25	23, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ 𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
26	22, 25	syl5bbr 274	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∈ 𝐽 → (¬ 𝑦 ∈ 𝑜 ↔ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
27	26	anbi2d 740	. . . . . . . . . . . . 13 ⊢ (𝑜 ∈ 𝐽 → ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
28	27	rexbiia 3040	. . . . . . . . . . . 12 ⊢ (∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
29		rexanali 2998	. . . . . . . . . . . 12 ⊢ (∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
30	28, 29	bitr3i 266	. . . . . . . . . . 11 ⊢ (∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) ↔ ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
31	30	con2bii 347	. . . . . . . . . 10 ⊢ (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
32	31	imbi1i 339	. . . . . . . . 9 ⊢ ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
33	19, 21, 32	3bitr4ri 293	. . . . . . . 8 ⊢ ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
34	33	imbi2i 326	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝑋 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))))
35		eldifsn 4317	. . . . . . . . 9 ⊢ (𝑥 ∈ (∪ 𝐽 ∖ {𝑦}) ↔ (𝑥 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦))
36	35	imbi1i 339	. . . . . . . 8 ⊢ ((𝑥 ∈ (∪ 𝐽 ∖ {𝑦}) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
37		impexp 462	. . . . . . . 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 264	. . . . . . 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 303	. . . . . 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 2984	. . . . 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 300	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
42	41	ralbidva 2985	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦 ∈ 𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
43		ralcom 3098	. . 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	syl6bb 276	. 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 296	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ∪ cuni 4436  ‘cfv 5888  Topctop 20698  TopOnctopon 20715  Clsdccld 20820  Frect1 21111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-topgen 16104  df-top 20699  df-topon 20716  df-cld 20823  df-t1 21118

This theorem is referenced by:  t1t0  21152  ist1-3  21153  haust1  21156  t1sep2  21173  isr0  21540  tgpt0  21922