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Theorem t1sncld 23350
Description: In a T1 space, singletons are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
ist0.1 𝑋 = 𝐽
Assertion
Ref Expression
t1sncld ((𝐽 ∈ Fre ∧ 𝐴𝑋) → {𝐴} ∈ (Clsd‘𝐽))

Proof of Theorem t1sncld
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ist0.1 . . . 4 𝑋 = 𝐽
21ist1 23345 . . 3 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽)))
3 sneq 4641 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43eleq1d 2824 . . . 4 (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽)))
54rspccv 3619 . . 3 (∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽) → (𝐴𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
62, 5simplbiim 504 . 2 (𝐽 ∈ Fre → (𝐴𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
76imp 406 1 ((𝐽 ∈ Fre ∧ 𝐴𝑋) → {𝐴} ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  {csn 4631   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040  Frect1 23331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-t1 23338
This theorem is referenced by:  cnt1  23374  lpcls  23388  sncld  23395  dnsconst  23402  t1connperf  23460  r0cld  23762  tgpt1  24142  sibfinima  34321  sibfof  34322
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