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Mirrors > Home > MPE Home > Th. List > reghaus | Structured version Visualization version GIF version |
Description: A regular T0 space is Hausdorff. In other words, a T3 space is T2 . A regular Hausdorff or T0 space is also known as a T3 space. (Contributed by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
reghaus | ⊢ (𝐽 ∈ Reg → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | haust1 21485 | . . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
2 | t1t0 21481 | . . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Kol2) |
4 | regr1 21882 | . . . . 5 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Haus) | |
5 | 4 | anim2i 611 | . . . 4 ⊢ ((𝐽 ∈ Kol2 ∧ 𝐽 ∈ Reg) → (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Haus)) |
6 | ishaus3 21955 | . . . 4 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Haus)) | |
7 | 5, 6 | sylibr 226 | . . 3 ⊢ ((𝐽 ∈ Kol2 ∧ 𝐽 ∈ Reg) → 𝐽 ∈ Haus) |
8 | 7 | expcom 403 | . 2 ⊢ (𝐽 ∈ Reg → (𝐽 ∈ Kol2 → 𝐽 ∈ Haus)) |
9 | 3, 8 | impbid2 218 | 1 ⊢ (𝐽 ∈ Reg → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∈ wcel 2157 ‘cfv 6101 Kol2ct0 21439 Frect1 21440 Hauscha 21441 Regcreg 21442 KQckq 21825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-1o 7799 df-map 8097 df-topgen 16419 df-qtop 16482 df-top 21027 df-topon 21044 df-cld 21152 df-cls 21154 df-cn 21360 df-t0 21446 df-t1 21447 df-haus 21448 df-reg 21449 df-kq 21826 df-hmeo 21887 df-hmph 21888 |
This theorem is referenced by: nrmhaus 21958 |
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