Theorem t0top 23249

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 2729	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ist0 23240	. 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 2109  ∀wral 3044  ∪ cuni 4867  Topctop 22813  Kol2ct0 23226

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-ss 3928  df-uni 4868  df-t0 23233

This theorem is referenced by:  restt0  23286  sst0  23293  kqt0  23666  t0hmph  23710  kqhmph  23739  ordtopt0  36423