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Mirrors > Home > MPE Home > Th. List > ishaus3 | Structured version Visualization version GIF version |
Description: A topological space is Hausdorff iff it is both T0 and R1 (where R1 means that any two topologically distinct points are separated by neighborhoods). (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
ishaus3 | ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Haus)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | haust1 21527 | . . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
2 | t1t0 21523 | . . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Kol2) |
4 | haushmph 21966 | . 2 ⊢ (𝐽 ≃ (KQ‘𝐽) → (𝐽 ∈ Haus → (KQ‘𝐽) ∈ Haus)) | |
5 | haushmph 21966 | . 2 ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ Haus → 𝐽 ∈ Haus)) | |
6 | 3, 4, 5 | ist1-5lem 21994 | 1 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Haus)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∈ wcel 2166 ‘cfv 6123 Kol2ct0 21481 Frect1 21482 Hauscha 21483 KQckq 21867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-1o 7826 df-map 8124 df-topgen 16457 df-qtop 16520 df-top 21069 df-topon 21086 df-cld 21194 df-cn 21402 df-t0 21488 df-t1 21489 df-haus 21490 df-kq 21868 df-hmeo 21929 df-hmph 21930 |
This theorem is referenced by: reghaus 21999 |
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