Theorem t1sncld 23688

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 23683	. . 3 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽)))
3		sneq 4594	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	eleq1d 2849	. . . 4 ⊢ (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽)))
5	4	rspccv 3577	. . 3 ⊢ (∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽) → (𝐴 ∈ 𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
6	2, 5	simplbiim 514	. 2 ⊢ (𝐽 ∈ Fre → (𝐴 ∈ 𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
7	6	imp 412	1 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ (Clsd‘𝐽))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {csn 4588  ∪ cuni 4870  ‘cfv 6564  Topctop 23255  Clsdccld 23378  Frect1 23669

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6518  df-fv 6572  df-t1 23676

This theorem is used by:  cnt1  23712  lpcls  23726  sncld  23733  dnsconst  23740  t1connperf  23798  r0cld  24101  tgpt1  24481  sibfinima  35075  sibfof  35076