Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  t1sncld Structured version   Visualization version   GIF version

Theorem t1sncld 20940
 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 . . . . 5 𝑋 = 𝐽
21ist1 20935 . . . 4 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽)))
32simprbi 479 . . 3 (𝐽 ∈ Fre → ∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽))
4 sneq 4135 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
54eleq1d 2672 . . . 4 (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽)))
65rspccv 3279 . . 3 (∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽) → (𝐴𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
73, 6syl 17 . 2 (𝐽 ∈ Fre → (𝐴𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
87imp 444 1 ((𝐽 ∈ Fre ∧ 𝐴𝑋) → {𝐴} ∈ (Clsd‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {csn 4125  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630  Frect1 20921 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-t1 20928 This theorem is referenced by:  cnt1  20964  lpcls  20978  sncld  20985  dnsconst  20992  t1conperf  21049  r0cld  21351  tgpt1  21731  sibfinima  29728  sibfof  29729
 Copyright terms: Public domain W3C validator