Theorem connclo 21218

Description: The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.)

Hypotheses

Ref	Expression
isconn.1	|- X = U. J
connclo.1	|- ( ph -> J e. Conn )
connclo.2	|- ( ph -> A e. J )
connclo.3	|- ( ph -> A =/= (/) )
connclo.4	|- ( ph -> A e. ( Clsd ` J ) )

Assertion

Ref	Expression
connclo	|- ( ph -> A = X )

Proof of Theorem connclo

Step	Hyp	Ref	Expression
1		connclo.3	. . 3 |- ( ph -> A =/= (/) )
2	1	neneqd 2799	. 2 |- ( ph -> -. A = (/) )
3		connclo.2	. . . . . 6 |- ( ph -> A e. J )
4		connclo.4	. . . . . 6 |- ( ph -> A e. ( Clsd ` J ) )
5	3, 4	elind 3798	. . . . 5 |- ( ph -> A e. ( J i^i ( Clsd ` J ) ) )
6		connclo.1	. . . . . 6 |- ( ph -> J e. Conn )
7		isconn.1	. . . . . . . 8 |- X = U. J
8	7	isconn 21216	. . . . . . 7 |- ( J e. Conn <-> ( J e. Top /\ ( J i^i ( Clsd ` J ) ) = { (/) , X } ) )
9	8	simprbi 480	. . . . . 6 |- ( J e. Conn -> ( J i^i ( Clsd ` J ) ) = { (/) , X } )
10	6, 9	syl 17	. . . . 5 |- ( ph -> ( J i^i ( Clsd ` J ) ) = { (/) , X } )
11	5, 10	eleqtrd 2703	. . . 4 |- ( ph -> A e. { (/) , X } )
12		elpri 4197	. . . 4 |- ( A e. { (/) , X } -> ( A = (/) \/ A = X ) )
13	11, 12	syl 17	. . 3 |- ( ph -> ( A = (/) \/ A = X ) )
14	13	ord 392	. 2 |- ( ph -> ( -. A = (/) -> A = X ) )
15	2, 14	mpd 15	1 |- ( ph -> A = X )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   = wceq 1483   e. wcel 1990   =/= wne 2794   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-ne 2795  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:  conndisj  21219  cnconn  21225  connsubclo  21227  t1connperf  21239  txconn  21492  connpconn  31217  cvmliftmolem2  31264  cvmlift2lem12  31296  mblfinlem1  33446