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Mirrors > Home > MPE Home > Th. List > nrmhaus | Structured version Visualization version GIF version |
Description: A T1 normal space is Hausdorff. A Hausdorff or T1 normal space is also known as a T4 space. (Contributed by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
nrmhaus | ⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | haust1 21960 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
2 | nrmreg 22432 | . . . 4 ⊢ ((𝐽 ∈ Nrm ∧ 𝐽 ∈ Fre) → 𝐽 ∈ Reg) | |
3 | 2 | ex 415 | . . 3 ⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Fre → 𝐽 ∈ Reg)) |
4 | t1t0 21956 | . . . 4 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2) | |
5 | reghaus 22433 | . . . 4 ⊢ (𝐽 ∈ Reg → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2)) | |
6 | 4, 5 | syl5ibrcom 249 | . . 3 ⊢ (𝐽 ∈ Fre → (𝐽 ∈ Reg → 𝐽 ∈ Haus)) |
7 | 3, 6 | sylcom 30 | . 2 ⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Fre → 𝐽 ∈ Haus)) |
8 | 1, 7 | impbid2 228 | 1 ⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 Kol2ct0 21914 Frect1 21915 Hauscha 21916 Regcreg 21917 Nrmcnrm 21918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-1o 8102 df-map 8408 df-topgen 16717 df-qtop 16780 df-top 21502 df-topon 21519 df-cld 21627 df-cls 21629 df-cn 21835 df-t0 21921 df-t1 21922 df-haus 21923 df-reg 21924 df-nrm 21925 df-kq 22302 df-hmeo 22363 df-hmph 22364 |
This theorem is referenced by: (None) |
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