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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 23199 | . . 3 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽))) |
3 | sneq 4634 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
4 | 3 | eleq1d 2813 | . . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽))) |
5 | 4 | rspccv 3604 | . . 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 1534 ∈ wcel 2099 ∀wral 3056 {csn 4624 ∪ cuni 4903 ‘cfv 6542 Topctop 22769 Clsdccld 22894 Frect1 23185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-t1 23192 |
This theorem is referenced by: cnt1 23228 lpcls 23242 sncld 23249 dnsconst 23256 t1connperf 23314 r0cld 23616 tgpt1 23996 sibfinima 33882 sibfof 33883 |
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