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Theorem t1sncld 21332
Description: In a T1 space, one-point sets 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 21327 . . 3 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽)))
3 sneq 4331 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43eleq1d 2824 . . . 4 (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽)))
54rspccv 3446 . . 3 (∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽) → (𝐴𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
62, 5simplbiim 661 . 2 (𝐽 ∈ Fre → (𝐴𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
76imp 444 1 ((𝐽 ∈ Fre ∧ 𝐴𝑋) → {𝐴} ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  {csn 4321   cuni 4588  cfv 6049  Topctop 20900  Clsdccld 21022  Frect1 21313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-t1 21320
This theorem is referenced by:  cnt1  21356  lpcls  21370  sncld  21377  dnsconst  21384  t1connperf  21441  r0cld  21743  tgpt1  22122  sibfinima  30710  sibfof  30711
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