Theorem conndisj 21219

Description: If a topology is connected, its underlying set can't be partitioned into two nonempty non-overlapping open sets. (Contributed by FL, 16-Nov-2008.) (Proof shortened 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 =/= (/) )
conndisj.4	|- ( ph -> B e. J )
conndisj.5	|- ( ph -> B =/= (/) )
conndisj.6	|- ( ph -> ( A i^i B ) = (/) )

Assertion

Ref	Expression
conndisj	|- ( ph -> ( A u. B ) =/= X )

Proof of Theorem conndisj

Step	Hyp	Ref	Expression
1		connclo.3	. 2 |- ( ph -> A =/= (/) )
2		connclo.2	. . . . . . 7 |- ( ph -> A e. J )
3		elssuni 4467	. . . . . . 7 |- ( A e. J -> A C_ U. J )
4	2, 3	syl 17	. . . . . 6 |- ( ph -> A C_ U. J )
5		isconn.1	. . . . . 6 |- X = U. J
6	4, 5	syl6sseqr 3652	. . . . 5 |- ( ph -> A C_ X )
7		conndisj.6	. . . . 5 |- ( ph -> ( A i^i B ) = (/) )
8		uneqdifeq 4057	. . . . 5 |- ( ( A C_ X /\ ( A i^i B ) = (/) ) -> ( ( A u. B ) = X <-> ( X \ A ) = B ) )
9	6, 7, 8	syl2anc 693	. . . 4 |- ( ph -> ( ( A u. B ) = X <-> ( X \ A ) = B ) )
10		simpr 477	. . . . . . 7 |- ( ( ph /\ ( X \ A ) = B ) -> ( X \ A ) = B )
11	10	difeq2d 3728	. . . . . 6 |- ( ( ph /\ ( X \ A ) = B ) -> ( X \ ( X \ A ) ) = ( X \ B ) )
12		dfss4 3858	. . . . . . . 8 |- ( A C_ X <-> ( X \ ( X \ A ) ) = A )
13	6, 12	sylib 208	. . . . . . 7 |- ( ph -> ( X \ ( X \ A ) ) = A )
14	13	adantr 481	. . . . . 6 |- ( ( ph /\ ( X \ A ) = B ) -> ( X \ ( X \ A ) ) = A )
15		connclo.1	. . . . . . . . . 10 |- ( ph -> J e. Conn )
16	15	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( X \ A ) = B ) -> J e. Conn )
17		conndisj.4	. . . . . . . . . 10 |- ( ph -> B e. J )
18	17	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( X \ A ) = B ) -> B e. J )
19		conndisj.5	. . . . . . . . . 10 |- ( ph -> B =/= (/) )
20	19	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( X \ A ) = B ) -> B =/= (/) )
21	5	isconn 21216	. . . . . . . . . . . . . 14 |- ( J e. Conn <-> ( J e. Top /\ ( J i^i ( Clsd ` J ) ) = { (/) , X } ) )
22	21	simplbi 476	. . . . . . . . . . . . 13 |- ( J e. Conn -> J e. Top )
23	15, 22	syl 17	. . . . . . . . . . . 12 |- ( ph -> J e. Top )
24	5	opncld 20837	. . . . . . . . . . . 12 |- ( ( J e. Top /\ A e. J ) -> ( X \ A ) e. ( Clsd ` J ) )
25	23, 2, 24	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( X \ A ) e. ( Clsd ` J ) )
26	25	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( X \ A ) = B ) -> ( X \ A ) e. ( Clsd ` J ) )
27	10, 26	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ph /\ ( X \ A ) = B ) -> B e. ( Clsd ` J ) )
28	5, 16, 18, 20, 27	connclo 21218	. . . . . . . 8 |- ( ( ph /\ ( X \ A ) = B ) -> B = X )
29	28	difeq2d 3728	. . . . . . 7 |- ( ( ph /\ ( X \ A ) = B ) -> ( X \ B ) = ( X \ X ) )
30		difid 3948	. . . . . . 7 |- ( X \ X ) = (/)
31	29, 30	syl6eq 2672	. . . . . 6 |- ( ( ph /\ ( X \ A ) = B ) -> ( X \ B ) = (/) )
32	11, 14, 31	3eqtr3d 2664	. . . . 5 |- ( ( ph /\ ( X \ A ) = B ) -> A = (/) )
33	32	ex 450	. . . 4 |- ( ph -> ( ( X \ A ) = B -> A = (/) ) )
34	9, 33	sylbid 230	. . 3 |- ( ph -> ( ( A u. B ) = X -> A = (/) ) )
35	34	necon3d 2815	. 2 |- ( ph -> ( A =/= (/) -> ( A u. B ) =/= X ) )
36	1, 35	mpd 15	1 |- ( ph -> ( A u. B ) =/= X )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   u. cun 3572   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-ne 2795  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:  dfconn2  21222