Theorem t0top 23271

Description: A T₀ space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Assertion

Ref	Expression
t0top	⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top)

Proof of Theorem t0top

Dummy variables 𝑥 𝑦 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2734	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ist0 23262	. 2 ⊢ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2113  ∀wral 3049  ∪ cuni 4861  Topctop 22835  Kol2ct0 23248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-ss 3916  df-uni 4862  df-t0 23255

This theorem is referenced by:  restt0  23308  sst0  23315  kqt0  23688  t0hmph  23732  kqhmph  23761  ordtopt0  36585