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Theorem t0top 21932
 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 2822 . . 3 𝐽 = 𝐽
21ist0 21923 . 2 (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
32simplbi 501 1 (𝐽 ∈ Kol2 → 𝐽 ∈ Top)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2114  ∀wral 3130  ∪ cuni 4813  Topctop 21496  Kol2ct0 21909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rab 3139  df-v 3471  df-in 3915  df-ss 3925  df-uni 4814  df-t0 21916 This theorem is referenced by:  restt0  21969  sst0  21976  kqt0  22349  t0hmph  22393  kqhmph  22422  ordtopt0  33864
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