MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  conndisj Unicode version

Theorem conndisj 17158
Description: If a topology is connected, its underlying set can't be partitioned into two non-empty non-overlapping open sets. (Contributed by FL, 16-Nov-2008.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
Hypotheses
Ref Expression
iscon.1  |-  X  = 
U. J
conclo.1  |-  ( ph  ->  J  e.  Con )
conclo.2  |-  ( ph  ->  A  e.  J )
conclo.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
StepHypRef Expression
1 conclo.3 . 2  |-  ( ph  ->  A  =/=  (/) )
2 conclo.2 . . . . . . 7  |-  ( ph  ->  A  e.  J )
3 elssuni 3871 . . . . . . 7  |-  ( A  e.  J  ->  A  C_ 
U. J )
42, 3syl 15 . . . . . 6  |-  ( ph  ->  A  C_  U. J )
5 iscon.1 . . . . . 6  |-  X  = 
U. J
64, 5syl6sseqr 3238 . . . . 5  |-  ( ph  ->  A  C_  X )
7 conndisj.6 . . . . 5  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
8 uneqdifeq 3555 . . . . 5  |-  ( ( A  C_  X  /\  ( A  i^i  B )  =  (/) )  ->  (
( A  u.  B
)  =  X  <->  ( X  \  A )  =  B ) )
96, 7, 8syl2anc 642 . . . 4  |-  ( ph  ->  ( ( A  u.  B )  =  X  <-> 
( X  \  A
)  =  B ) )
10 simpr 447 . . . . . . 7  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  A )  =  B )
1110difeq2d 3307 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  ( X  \  A
) )  =  ( X  \  B ) )
12 dfss4 3416 . . . . . . . 8  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
136, 12sylib 188 . . . . . . 7  |-  ( ph  ->  ( X  \  ( X  \  A ) )  =  A )
1413adantr 451 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  ( X  \  A
) )  =  A )
15 conclo.1 . . . . . . . . . 10  |-  ( ph  ->  J  e.  Con )
1615adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  J  e.  Con )
17 conndisj.4 . . . . . . . . . 10  |-  ( ph  ->  B  e.  J )
1817adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  e.  J )
19 conndisj.5 . . . . . . . . . 10  |-  ( ph  ->  B  =/=  (/) )
2019adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  =/=  (/) )
215iscon 17155 . . . . . . . . . . . . . 14  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
2221simplbi 446 . . . . . . . . . . . . 13  |-  ( J  e.  Con  ->  J  e.  Top )
2315, 22syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  Top )
245opncld 16786 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  ( X  \  A
)  e.  ( Clsd `  J ) )
2523, 2, 24syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( X  \  A
)  e.  ( Clsd `  J ) )
2625adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  A )  e.  (
Clsd `  J )
)
2710, 26eqeltrrd 2371 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  e.  ( Clsd `  J )
)
285, 16, 18, 20, 27conclo 17157 . . . . . . . 8  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  =  X )
2928difeq2d 3307 . . . . . . 7  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  B )  =  ( X  \  X ) )
30 difid 3535 . . . . . . 7  |-  ( X 
\  X )  =  (/)
3129, 30syl6eq 2344 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  B )  =  (/) )
3211, 14, 313eqtr3d 2336 . . . . 5  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  A  =  (/) )
3332ex 423 . . . 4  |-  ( ph  ->  ( ( X  \  A )  =  B  ->  A  =  (/) ) )
349, 33sylbid 206 . . 3  |-  ( ph  ->  ( ( A  u.  B )  =  X  ->  A  =  (/) ) )
3534necon3d 2497 . 2  |-  ( ph  ->  ( A  =/=  (/)  ->  ( A  u.  B )  =/=  X ) )
361, 35mpd 14 1  |-  ( ph  ->  ( A  u.  B
)  =/=  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {cpr 3654   U.cuni 3843   ` cfv 5271   Topctop 16647   Clsdccld 16769   Conccon 17153
This theorem is referenced by:  dfcon2  17161  usinuniopb  25697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-top 16652  df-cld 16772  df-con 17154
  Copyright terms: Public domain W3C validator