Theorem t1sncld 23334

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 23329	. . 3 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽)))
3		sneq 4636	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	eleq1d 2826	. . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽)))
5	4	rspccv 3619	. . 3 ⊢ (∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽) → (𝐴 ∈ 𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
6	2, 5	simplbiim 504	. 2 ⊢ (𝐽 ∈ Fre → (𝐴 ∈ 𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
7	6	imp 406	1 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {csn 4626  ∪ cuni 4907  ‘cfv 6561  Topctop 22899  Clsdccld 23024  Frect1 23315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-t1 23322

This theorem is referenced by:  cnt1  23358  lpcls  23372  sncld  23379  dnsconst  23386  t1connperf  23444  r0cld  23746  tgpt1  24126  sibfinima  34341  sibfof  34342