Theorem t1sncld 21931

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 ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ (Clsd‘𝐽))

Proof of Theorem t1sncld

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ist0.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	ist1 21926	. . 3 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽)))
3		sneq 4535	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	eleq1d 2874	. . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽)))
5	4	rspccv 3568	. . 3 ⊢ (∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽) → (𝐴 ∈ 𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
6	2, 5	simplbiim 508	. 2 ⊢ (𝐽 ∈ Fre → (𝐴 ∈ 𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
7	6	imp 410	1 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {csn 4525  ∪ cuni 4800  ‘cfv 6324  Topctop 21498  Clsdccld 21621  Frect1 21912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-t1 21919

This theorem is referenced by:  cnt1  21955  lpcls  21969  sncld  21976  dnsconst  21983  t1connperf  22041  r0cld  22343  tgpt1  22723  sibfinima  31707  sibfof  31708