Theorem t0top 22480

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 2738	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ist0 22471	. 2 ⊢ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
3	2	simplbi 498	1 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2106  ∀wral 3064  ∪ cuni 4839  Topctop 22042  Kol2ct0 22457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-t0 22464

This theorem is referenced by:  restt0  22517  sst0  22524  kqt0  22897  t0hmph  22941  kqhmph  22970  ordtopt0  34631