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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 23329 | . . 3 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽))) |
| 3 | sneq 4636 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
| 4 | 3 | eleq1d 2826 | . . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽))) |
| 5 | 4 | rspccv 3619 | . . 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 1540 ∈ wcel 2108 ∀wral 3061 {csn 4626 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 Clsdccld 23024 Frect1 23315 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-t1 23322 |
| This theorem is referenced by: cnt1 23358 lpcls 23372 sncld 23379 dnsconst 23386 t1connperf 23444 r0cld 23746 tgpt1 24126 sibfinima 34341 sibfof 34342 |
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