MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  t0top Structured version   Visualization version   GIF version

Theorem t0top 22703
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 2733 . . 3 𝐽 = 𝐽
21ist0 22694 . 2 (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
32simplbi 499 1 (𝐽 ∈ Kol2 → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107  wral 3061   cuni 4869  Topctop 22265  Kol2ct0 22680
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3449  df-in 3921  df-ss 3931  df-uni 4870  df-t0 22687
This theorem is referenced by:  restt0  22740  sst0  22747  kqt0  23120  t0hmph  23164  kqhmph  23193  ordtopt0  34967
  Copyright terms: Public domain W3C validator