Theorem haust1 23076

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 2732	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	hausnei 23052	. . . . . . . 8 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
3		simprr1 1221	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → 𝑥 ∈ 𝑧)
4		noel 4330	. . . . . . . . . . . . 13 ⊢ ¬ 𝑦 ∈ ∅
5		simprr3 1223	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → (𝑧 ∩ 𝑤) = ∅)
6	5	eleq2d 2819	. . . . . . . . . . . . 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 326	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → ¬ 𝑦 ∈ (𝑧 ∩ 𝑤))
8		simprr2 1222	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → 𝑦 ∈ 𝑤)
9		elin 3964	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝑧 ∩ 𝑤) ↔ (𝑦 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤))
10	9	simplbi2com 503	. . . . . . . . . . . . 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 197	. . . . . . . . . . 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 512	. . . . . . . . . 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 3156	. . . . . . . . 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 3168	. . . . . . . 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 3102	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ¬ 𝑦 ∈ 𝑧) ↔ ¬ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
18	16, 17	sylib 217	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ¬ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
19	18	3exp2 1354	. . . . 5 ⊢ (𝐽 ∈ Haus → (𝑥 ∈ ∪ 𝐽 → (𝑦 ∈ ∪ 𝐽 → (𝑥 ≠ 𝑦 → ¬ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))))
20	19	imp32 419	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽)) → (𝑥 ≠ 𝑦 → ¬ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
21	20	necon4ad 2959	. . 3 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽)) → (∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦))
22	21	ralrimivva 3200	. 2 ⊢ (𝐽 ∈ Haus → ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦))
23		haustop 23055	. . . 4 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
24		toptopon2 22640	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
25	23, 24	sylib 217	. . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘∪ 𝐽))
26		ist1-2 23071	. . 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 256	1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∩ cin 3947  ∅c0 4322  ∪ cuni 4908  ‘cfv 6543  Topctop 22615  TopOnctopon 22632  Frect1 23031  Hauscha 23032

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-topgen 17393  df-top 22616  df-topon 22633  df-cld 22743  df-t1 23038  df-haus 23039

This theorem is referenced by:  sncld  23095  ishaus3  23547  reghaus  23549  nrmhaus  23550  tgpt1  23842  metreg  24599  ipasslem8  30345  sitmcl  33636  onint1  35637  oninhaus  35638  poimirlem30  36821