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Mirrors > Home > MPE Home > Th. List > sncld | Structured version Visualization version GIF version |
Description: A singleton is closed in a Hausdorff space. (Contributed by NM, 5-Mar-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
t1sep.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
sncld | ⊢ ((𝐽 ∈ Haus ∧ 𝑃 ∈ 𝑋) → {𝑃} ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | haust1 21960 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
2 | t1sep.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | t1sncld 21934 | . 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑃 ∈ 𝑋) → {𝑃} ∈ (Clsd‘𝐽)) |
4 | 1, 3 | sylan 582 | 1 ⊢ ((𝐽 ∈ Haus ∧ 𝑃 ∈ 𝑋) → {𝑃} ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4567 ∪ cuni 4838 ‘cfv 6355 Clsdccld 21624 Frect1 21915 Hauscha 21916 |
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-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-rab 3147 df-v 3496 df-sbc 3773 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-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-iota 6314 df-fun 6357 df-fv 6363 df-topgen 16717 df-top 21502 df-topon 21519 df-cld 21627 df-t1 21922 df-haus 21923 |
This theorem is referenced by: tgphaus 22725 csscld 23852 clsocv 23853 dvrec 24552 dvexp3 24575 abelth 25029 dvtanlem 34956 sncldre 41324 dirkercncflem2 42409 |
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