Theorem haust1 23317

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 2736	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	hausnei 23293	. . . . . . . 8 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
3		simprr1 1223	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → 𝑥 ∈ 𝑧)
4		noel 4278	. . . . . . . . . . . . 13 ⊢ ¬ 𝑦 ∈ ∅
5		simprr3 1225	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → (𝑧 ∩ 𝑤) = ∅)
6	5	eleq2d 2822	. . . . . . . . . . . . 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 1224	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → 𝑦 ∈ 𝑤)
9		elin 3905	. . . . . . . . . . . . . 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 3139	. . . . . . . . 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 3150	. . . . . . . 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 3091	. . . . . . 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 1356	. . . . 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 2951	. . 3 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽)) → (∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦))
22	21	ralrimivva 3180	. 2 ⊢ (𝐽 ∈ Haus → ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦))
23		haustop 23296	. . . 4 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
24		toptopon2 22883	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
25	23, 24	sylib 218	. . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘∪ 𝐽))
26		ist1-2 23312	. . 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 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061   ∩ cin 3888  ∅c0 4273  ∪ cuni 4850  ‘cfv 6498  Topctop 22858  TopOnctopon 22875  Frect1 23272  Hauscha 23273

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-topgen 17406  df-top 22859  df-topon 22876  df-cld 22984  df-t1 23279  df-haus 23280

This theorem is referenced by:  sncld  23336  ishaus3  23788  reghaus  23790  nrmhaus  23791  tgpt1  24083  metreg  24829  ipasslem8  30908  sitmcl  34495  onint1  36631  oninhaus  36632  poimirlem30  37971