Theorem ist1-2 23376

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 22940	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		eqid 2740	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	ist1 23350	. . . 4 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽)))
4	3	baib 535	. . 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 22941	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	raleqdv 3334	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦 ∈ 𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 ∈ ∪ 𝐽{𝑦} ∈ (Clsd‘𝐽)))
8	1	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐽 ∈ Top)
9		eltop2 23003	. . . . . 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 2830	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝐽))
12	11	biimpa 476	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ∪ 𝐽)
13	12	snssd 4834	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → {𝑦} ⊆ ∪ 𝐽)
14	2	iscld2 23057	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝑦} ⊆ ∪ 𝐽) → ({𝑦} ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ {𝑦}) ∈ 𝐽))
15	8, 13, 14	syl2anc 583	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ {𝑦}) ∈ 𝐽))
16	6	adantr 480	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
17	16	eleq2d 2830	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽))
18	17	imbi1d 341	. . . . . . 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 358	. . . . . . . . 9 ⊢ ((¬ 𝑥 = 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
20		df-ne 2947	. . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
21	20	imbi1i 349	. . . . . . . . 9 ⊢ ((𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ (¬ 𝑥 = 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
22		disjsn 4736	. . . . . . . . . . . . . . 15 ⊢ ((𝑜 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑜)
23		elssuni 4961	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
24		reldisj 4476	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ⊆ ∪ 𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
25	23, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ 𝐽 → ((𝑜 ∩ {𝑦}) = ∅ ↔ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
26	22, 25	bitr3id 285	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∈ 𝐽 → (¬ 𝑦 ∈ 𝑜 ↔ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
27	26	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑜 ∈ 𝐽 → ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
28	27	rexbiia 3098	. . . . . . . . . . . 12 ⊢ (∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
29		rexanali 3108	. . . . . . . . . . . 12 ⊢ (∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
30	28, 29	bitr3i 277	. . . . . . . . . . 11 ⊢ (∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) ↔ ¬ ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
31	30	con2bii 357	. . . . . . . . . 10 ⊢ (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
32	31	imbi1i 349	. . . . . . . . 9 ⊢ ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (¬ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})) → 𝑥 = 𝑦))
33	19, 21, 32	3bitr4ri 304	. . . . . . . 8 ⊢ ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
34	33	imbi2i 336	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝑋 → (𝑥 ≠ 𝑦 → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))))
35		eldifsn 4811	. . . . . . . . 9 ⊢ (𝑥 ∈ (∪ 𝐽 ∖ {𝑦}) ↔ (𝑥 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦))
36	35	imbi1i 349	. . . . . . . 8 ⊢ ((𝑥 ∈ (∪ 𝐽 ∖ {𝑦}) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))) ↔ ((𝑥 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
37		impexp 450	. . . . . . . 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 275	. . . . . . 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 314	. . . . . 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 3180	. . . . 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 311	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → ({𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
42	41	ralbidva 3182	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦 ∈ 𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
43		ralcom 3295	. . 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 287	. 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 307	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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648  ∪ cuni 4931  ‘cfv 6573  Topctop 22920  TopOnctopon 22937  Clsdccld 23045  Frect1 23336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-topgen 17503  df-top 22921  df-topon 22938  df-cld 23048  df-t1 23343

This theorem is referenced by:  t1t0  23377  ist1-3  23378  haust1  23381  t1sep2  23398  isr0  23766  tgpt0  24148