Theorem conntop 21220

Description: A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)

Assertion

Ref	Expression
conntop	|- ( J e. Conn -> J e. Top )

Proof of Theorem conntop

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- U. J = U. J
2	1	isconn 21216	. 2 |- ( J e. Conn <-> ( J e. Top /\ ( J i^i ( Clsd ` J ) ) = { (/) , U. J } ) )
3	2	simplbi 476	1 |- ( J e. Conn -> J e. Top )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   i^i cin 3573   (/)c0 3915   {cpr 4179   U.cuni 4436   ` cfv 5888   Topctop 20698   Clsdccld 20820  Conncconn 21214

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-3an 1039  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-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-conn 21215

This theorem is referenced by:  conncompss  21236  txconn  21492  qtopconn  21512  ufildr  21735  connpconn  31217  cvmliftmolem1  31263  cvmliftmolem2  31264  ordtopconn  32438