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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 23206 | . . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
2 | t1t0 23202 | . . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Kol2) |
4 | haushmph 23646 | . 2 ⊢ (𝐽 ≃ (KQ‘𝐽) → (𝐽 ∈ Haus → (KQ‘𝐽) ∈ Haus)) | |
5 | haushmph 23646 | . 2 ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ Haus → 𝐽 ∈ Haus)) | |
6 | 3, 4, 5 | ist1-5lem 23674 | 1 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Haus)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ‘cfv 6536 Kol2ct0 23160 Frect1 23161 Hauscha 23162 KQckq 23547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-1o 8464 df-map 8821 df-topgen 17395 df-qtop 17459 df-top 22746 df-topon 22763 df-cld 22873 df-cn 23081 df-t0 23167 df-t1 23168 df-haus 23169 df-kq 23548 df-hmeo 23609 df-hmph 23610 |
This theorem is referenced by: reghaus 23679 |
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