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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 20966 | . . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
2 | t1t0 20962 | . . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Kol2) |
4 | regr1 21363 | . . . . 5 ⊢ (𝐽 ∈ Reg → (KQ‘𝐽) ∈ Haus) | |
5 | 4 | anim2i 591 | . . . 4 ⊢ ((𝐽 ∈ Kol2 ∧ 𝐽 ∈ Reg) → (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Haus)) |
6 | ishaus3 21436 | . . . 4 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Haus)) | |
7 | 5, 6 | sylibr 223 | . . 3 ⊢ ((𝐽 ∈ Kol2 ∧ 𝐽 ∈ Reg) → 𝐽 ∈ Haus) |
8 | 7 | expcom 450 | . 2 ⊢ (𝐽 ∈ Reg → (𝐽 ∈ Kol2 → 𝐽 ∈ Haus)) |
9 | 3, 8 | impbid2 215 | 1 ⊢ (𝐽 ∈ Reg → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ‘cfv 5804 Kol2ct0 20920 Frect1 20921 Hauscha 20922 Regcreg 20923 KQckq 21306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-1o 7447 df-map 7746 df-topgen 15927 df-qtop 15990 df-top 20521 df-topon 20523 df-cld 20633 df-cls 20635 df-cn 20841 df-t0 20927 df-t1 20928 df-haus 20929 df-reg 20930 df-kq 21307 df-hmeo 21368 df-hmph 21369 |
This theorem is referenced by: nrmhaus 21439 |
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