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Theorem t0top 20943
 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.
StepHypRef Expression
1 eqid 2610 . . 3 𝐽 = 𝐽
21ist0 20934 . 2 (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
32simplbi 475 1 (𝐽 ∈ Kol2 → 𝐽 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  ∀wral 2896  ∪ cuni 4372  Topctop 20517  Kol2ct0 20920 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-uni 4373  df-t0 20927 This theorem is referenced by:  restt0  20980  sst0  20987  kqt0  21359  t0hmph  21403  kqhmph  21432  ordtopt0  31611
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