Theorem haust1 21644

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 2795	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	hausnei 21620	. . . . . . . 8 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
3		simprr1 1214	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → 𝑥 ∈ 𝑧)
4		noel 4216	. . . . . . . . . . . . 13 ⊢ ¬ 𝑦 ∈ ∅
5		simprr3 1216	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → (𝑧 ∩ 𝑤) = ∅)
6	5	eleq2d 2868	. . . . . . . . . . . . 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 328	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → ¬ 𝑦 ∈ (𝑧 ∩ 𝑤))
8		simprr2 1215	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) ∧ 𝑧 ∈ 𝐽) ∧ (𝑤 ∈ 𝐽 ∧ (𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) → 𝑦 ∈ 𝑤)
9		elin 4090	. . . . . . . . . . . . . 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 199	. . . . . . . . . . 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 3248	. . . . . . . . 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 3237	. . . . . . . 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 3229	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ ¬ 𝑦 ∈ 𝑧) ↔ ¬ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
18	16, 17	sylib 219	. . . . . 6 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ¬ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
19	18	3exp2 1347	. . . . 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 3003	. . 3 ⊢ ((𝐽 ∈ Haus ∧ (𝑥 ∈ ∪ 𝐽 ∧ 𝑦 ∈ ∪ 𝐽)) → (∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦))
22	21	ralrimivva 3158	. 2 ⊢ (𝐽 ∈ Haus → ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦))
23		haustop 21623	. . . 4 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
24		toptopon2 21210	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
25	23, 24	sylib 219	. . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ (TopOn‘∪ 𝐽))
26		ist1-2 21639	. . 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 258	1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  ∀wral 3105  ∃wrex 3106   ∩ cin 3858  ∅c0 4211  ∪ cuni 4745  ‘cfv 6225  Topctop 21185  TopOnctopon 21202  Frect1 21599  Hauscha 21600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-iota 6189  df-fun 6227  df-fv 6233  df-topgen 16546  df-top 21186  df-topon 21203  df-cld 21311  df-t1 21606  df-haus 21607

This theorem is referenced by:  sncld  21663  ishaus3  22115  reghaus  22117  nrmhaus  22118  tgpt1  22409  metreg  23154  ipasslem8  28305  sitmcl  31226  onint1  33407  oninhaus  33408  poimirlem30  34472