Theorem t0top 21937

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 21928	. 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 2114  ∀wral 3138  ∪ cuni 4838  Topctop 21501  Kol2ct0 21914

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-in 3943  df-ss 3952  df-uni 4839  df-t0 21921

This theorem is referenced by:  restt0  21974  sst0  21981  kqt0  22354  t0hmph  22398  kqhmph  22427  ordtopt0  33790