Theorem isconn2 21217

Description: The predicate J is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.)

Hypothesis

Ref	Expression
isconn.1	|- X = U. J

Assertion

Ref	Expression
isconn2	|- ( J e. Conn <-> ( J e. Top /\ ( J i^i ( Clsd ` J ) ) C_ { (/) , X } ) )

Proof of Theorem isconn2

Step	Hyp	Ref	Expression
1		isconn.1	. . 3 |- X = U. J
2	1	isconn 21216	. 2 |- ( J e. Conn <-> ( J e. Top /\ ( J i^i ( Clsd ` J ) ) = { (/) , X } ) )
3		0opn 20709	. . . . . . 7 |- ( J e. Top -> (/) e. J )
4		0cld 20842	. . . . . . 7 |- ( J e. Top -> (/) e. ( Clsd ` J ) )
5	3, 4	elind 3798	. . . . . 6 |- ( J e. Top -> (/) e. ( J i^i ( Clsd ` J ) ) )
6	1	topopn 20711	. . . . . . 7 |- ( J e. Top -> X e. J )
7	1	topcld 20839	. . . . . . 7 |- ( J e. Top -> X e. ( Clsd ` J ) )
8	6, 7	elind 3798	. . . . . 6 |- ( J e. Top -> X e. ( J i^i ( Clsd ` J ) ) )
9		prssi 4353	. . . . . 6 |- ( ( (/) e. ( J i^i ( Clsd ` J ) ) /\ X e. ( J i^i ( Clsd ` J ) ) ) -> { (/) , X } C_ ( J i^i ( Clsd ` J ) ) )
10	5, 8, 9	syl2anc 693	. . . . 5 |- ( J e. Top -> { (/) , X } C_ ( J i^i ( Clsd ` J ) ) )
11	10	biantrud 528	. . . 4 |- ( J e. Top -> ( ( J i^i ( Clsd ` J ) ) C_ { (/) , X } <-> ( ( J i^i ( Clsd ` J ) ) C_ { (/) , X } /\ { (/) , X } C_ ( J i^i ( Clsd ` J ) ) ) ) )
12		eqss 3618	. . . 4 |- ( ( J i^i ( Clsd ` J ) ) = { (/) , X } <-> ( ( J i^i ( Clsd ` J ) ) C_ { (/) , X } /\ { (/) , X } C_ ( J i^i ( Clsd ` J ) ) ) )
13	11, 12	syl6rbbr 279	. . 3 |- ( J e. Top -> ( ( J i^i ( Clsd ` J ) ) = { (/) , X } <-> ( J i^i ( Clsd ` J ) ) C_ { (/) , X } ) )
14	13	pm5.32i 669	. 2 |- ( ( J e. Top /\ ( J i^i ( Clsd ` J ) ) = { (/) , X } ) <-> ( J e. Top /\ ( J i^i ( Clsd ` J ) ) C_ { (/) , X } ) )
15	2, 14	bitri 264	1 |- ( J e. Conn <-> ( J e. Top /\ ( J i^i ( Clsd ` J ) ) C_ { (/) , X } ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   (/)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  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-eu 2474  df-mo 2475  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-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-top 20699  df-cld 20823  df-conn 21215

This theorem is referenced by:  indisconn  21221  dfconn2  21222  cnconn  21225  txconn  21492  filconn  21687  onsucconni  32436