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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 23175 | . . 3 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽))) |
3 | sneq 4633 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
4 | 3 | eleq1d 2812 | . . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽))) |
5 | 4 | rspccv 3603 | . . 3 ⊢ (∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽) → (𝐴 ∈ 𝑋 → {𝐴} ∈ (Clsd‘𝐽))) |
6 | 2, 5 | simplbiim 504 | . 2 ⊢ (𝐽 ∈ Fre → (𝐴 ∈ 𝑋 → {𝐴} ∈ (Clsd‘𝐽))) |
7 | 6 | imp 406 | 1 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 {csn 4623 ∪ cuni 4902 ‘cfv 6536 Topctop 22745 Clsdccld 22870 Frect1 23161 |
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 2697 |
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-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6488 df-fv 6544 df-t1 23168 |
This theorem is referenced by: cnt1 23204 lpcls 23218 sncld 23225 dnsconst 23232 t1connperf 23290 r0cld 23592 tgpt1 23972 sibfinima 33867 sibfof 33868 |
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