Theorem t0top 21133

Description: A T₀ space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Assertion

Ref	Expression
t0top	|- ( J e. Kol2 -> J e. 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 2622	. . 3 |- U. J = U. J
2	1	ist0 21124	. 2 |- ( J e. Kol2 <-> ( J e. Top /\ A. x e. U. J A. y e. U. J ( A. o e. J ( x e. o <-> y e. o ) -> x = y ) ) )
3	2	simplbi 476	1 |- ( J e. Kol2 -> J e. Top )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   A.wral 2912   U.cuni 4436   Topctop 20698   Kol2ct0 21110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-uni 4437  df-t0 21117

This theorem is referenced by:  restt0  21170  sst0  21177  kqt0  21549  t0hmph  21593  kqhmph  21622  ordtopt0  32441