Theorem haust1 23294

Description: A Hausdorff space is a T₁ space. (Contributed by FL, 11-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
haust1	⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre)

Proof of Theorem haust1

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2734	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	hausnei 23270	. . . . . . . 8 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
3		simprr1 1222	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → 𝑥 ∈ 𝑧)
4		noel 4288	. . . . . . . . . . . . 13 ⊢ ¬ 𝑦 ∈ ∅
5		simprr3 1224	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → (𝑧 ∩ 𝑤) = ∅)
6	5	eleq2d 2820	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → (𝑦 ∈ (𝑧 ∩ 𝑤) ↔ 𝑦 ∈ ∅))
7	4, 6	mtbiri 327	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → ¬ 𝑦 ∈ (𝑧 ∩ 𝑤))
8		simprr2 1223	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → 𝑦 ∈ 𝑤)
9		elin 3915	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝑧 ∩ 𝑤) ↔ (𝑦 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤))
10	9	simplbi2com 502	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑤 → (𝑦 ∈ 𝑧 → 𝑦 ∈ (𝑧 ∩ 𝑤)))
11	8, 10	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → (𝑦 ∈ 𝑧 → 𝑦 ∈ (𝑧 ∩ 𝑤)))
12	7, 11	mtod 198	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → ¬ 𝑦 ∈ 𝑧)
13	3, 12	jca 511	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → (𝑥 ∈ 𝑧 ∧ ¬ 𝑦 ∈ 𝑧))
14	13	rexlimdvaa 3136	. . . . . . . . 9 ⊢ (((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) → (∃𝑤 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅) → (𝑥 ∈ 𝑧 ∧ ¬ 𝑦 ∈ 𝑧)))
15	14	reximdva 3147	. . . . . . . 8 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ¬ 𝑦 ∈ 𝑧)))
16	2, 15	mpd 15	. . . . . . 7 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ¬ 𝑦 ∈ 𝑧))
17		rexanali 3088	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ¬ 𝑦 ∈ 𝑧) ↔ ¬ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
18	16, 17	sylib 218	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ¬ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
19	18	3exp2 1355	. . . . 5 ⊢ (𝐽 ∈ Haus → (𝑥 ∈ ∪ 𝐽 → (𝑦 ∈ ∪ 𝐽 → (𝑥 ≠ 𝑦 → ¬ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))))
20	19	imp32 418	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽)) → (𝑥 ≠ 𝑦 → ¬ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
21	20	necon4ad 2949	. . 3 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽)) → (∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦))
22	21	ralrimivva 3177	. 2 ⊢ (𝐽 ∈ Haus → ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦))
23		haustop 23273	. . . 4 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
24		toptopon2 22860	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
25	23, 24	sylib 218	. . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘∪ 𝐽))
26		ist1-2 23289	. . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦)))
27	25, 26	syl 17	. 2 ⊢ (𝐽 ∈ Haus → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦)))
28	22, 27	mpbird 257	1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058   ∩ cin 3898  ∅c0 4283  ∪ cuni 4861  ‘cfv 6490  Topctop 22835  TopOnctopon 22852  Frect1 23249  Hauscha 23250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-topgen 17361  df-top 22836  df-topon 22853  df-cld 22961  df-t1 23256  df-haus 23257

This theorem is referenced by:  sncld  23313  ishaus3  23765  reghaus  23767  nrmhaus  23768  tgpt1  24060  metreg  24806  ipasslem8  30861  sitmcl  34457  onint1  36592  oninhaus  36593  poimirlem30  37790