Theorem ist1-2 23322

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 22888	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		eqid 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	ist1 23296	. . . 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 22889	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	raleqdv 3296	. 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 22950	. . . . . 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 2823	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝐽))
12	11	biimpa 476	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ∪ 𝐽)
13	12	snssd 4753	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → {𝑦} ⊆ ∪ 𝐽)
14	2	iscld2 23003	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝑦} ⊆ ∪ 𝐽) → ({𝑦} ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ {𝑦}) ∈ 𝐽))
15	8, 13, 14	syl2anc 585	. . . . 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 2823	. . . . . . . 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 2934	. . . . . . . . . 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 4656	. . . . . . . . . . . . . . 15 ⊢ ((𝑜 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑜)
23		elssuni 4882	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽)
24		reldisj 4394	. . . . . . . . . . . . . . . 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 631	. . . . . . . . . . . . 13 ⊢ (𝑜 ∈ 𝐽 → ((𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦}))))
28	27	rexbiia 3083	. . . . . . . . . . . 12 ⊢ (∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ ¬ 𝑦 ∈ 𝑜) ↔ ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ 𝑜 ⊆ (∪ 𝐽 ∖ {𝑦})))
29		rexanali 3092	. . . . . . . . . . . 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 4730	. . . . . . . . 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 3157	. . . . 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 3159	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑦 ∈ 𝑋 {𝑦} ∈ (Clsd‘𝐽) ↔ ∀𝑦 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
43		ralcom 3266	. . 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 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  {csn 4568  ∪ cuni 4851  ‘cfv 6492  Topctop 22868  TopOnctopon 22885  Clsdccld 22991  Frect1 23282

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-topgen 17397  df-top 22869  df-topon 22886  df-cld 22994  df-t1 23289

This theorem is referenced by:  t1t0  23323  ist1-3  23324  haust1  23327  t1sep2  23344  isr0  23712  tgpt0  24094