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Mirrors > Home > MPE Home > Th. List > t1sncld | Structured version Visualization version GIF version |
Description: In a T1 space, singletons are closed. (Contributed by Jeff Hankins, 1-Feb-2010.) |
Ref | Expression |
---|---|
ist0.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
t1sncld | ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ist0.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | ist1 23245 | . . 3 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽))) |
3 | sneq 4642 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
4 | 3 | eleq1d 2814 | . . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽))) |
5 | 4 | rspccv 3608 | . . 3 ⊢ (∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽) → (𝐴 ∈ 𝑋 → {𝐴} ∈ (Clsd‘𝐽))) |
6 | 2, 5 | simplbiim 503 | . 2 ⊢ (𝐽 ∈ Fre → (𝐴 ∈ 𝑋 → {𝐴} ∈ (Clsd‘𝐽))) |
7 | 6 | imp 405 | 1 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 {csn 4632 ∪ cuni 4912 ‘cfv 6553 Topctop 22815 Clsdccld 22940 Frect1 23231 |
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-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-t1 23238 |
This theorem is referenced by: cnt1 23274 lpcls 23288 sncld 23295 dnsconst 23302 t1connperf 23360 r0cld 23662 tgpt1 24042 sibfinima 33992 sibfof 33993 |
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