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Theorem t1sncld 21926
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 21921 . . 3 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽)))
3 sneq 4569 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43eleq1d 2895 . . . 4 (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽)))
54rspccv 3618 . . 3 (∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽) → (𝐴𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
62, 5simplbiim 507 . 2 (𝐽 ∈ Fre → (𝐴𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
76imp 409 1 ((𝐽 ∈ Fre ∧ 𝐴𝑋) → {𝐴} ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  wral 3136  {csn 4559   cuni 4830  cfv 6348  Topctop 21493  Clsdccld 21616  Frect1 21907
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-t1 21914
This theorem is referenced by:  cnt1  21950  lpcls  21964  sncld  21971  dnsconst  21978  t1connperf  22036  r0cld  22338  tgpt1  22718  sibfinima  31590  sibfof  31591
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