Theorem t0top 21931

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 2821	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	ist0 21922	. 2 ⊢ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
3	2	simplbi 500	1 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2110  ∀wral 3138  ∪ cuni 4831  Topctop 21495  Kol2ct0 21908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-uni 4832  df-t0 21915

This theorem is referenced by:  restt0  21968  sst0  21975  kqt0  22348  t0hmph  22392  kqhmph  22421  ordtopt0  33785