Theorem t1sncld 23355

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 23350	. . 3 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽)))
3		sneq 4658	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	eleq1d 2829	. . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽)))
5	4	rspccv 3632	. . 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 1537   ∈ wcel 2108  ∀wral 3067  {csn 4648  ∪ cuni 4931  ‘cfv 6573  Topctop 22920  Clsdccld 23045  Frect1 23336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-t1 23343

This theorem is referenced by:  cnt1  23379  lpcls  23393  sncld  23400  dnsconst  23407  t1connperf  23465  r0cld  23767  tgpt1  24147  sibfinima  34304  sibfof  34305