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Theorem conndisj 17480
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 4044 . . . . . . 7  |-  ( A  e.  J  ->  A  C_ 
U. J )
42, 3syl 16 . . . . . 6  |-  ( ph  ->  A  C_  U. J )
5 iscon.1 . . . . . 6  |-  X  = 
U. J
64, 5syl6sseqr 3396 . . . . 5  |-  ( ph  ->  A  C_  X )
7 conndisj.6 . . . . 5  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
8 uneqdifeq 3717 . . . . 5  |-  ( ( A  C_  X  /\  ( A  i^i  B )  =  (/) )  ->  (
( A  u.  B
)  =  X  <->  ( X  \  A )  =  B ) )
96, 7, 8syl2anc 644 . . . 4  |-  ( ph  ->  ( ( A  u.  B )  =  X  <-> 
( X  \  A
)  =  B ) )
10 simpr 449 . . . . . . 7  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  A )  =  B )
1110difeq2d 3466 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  ( X  \  A
) )  =  ( X  \  B ) )
12 dfss4 3576 . . . . . . . 8  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
136, 12sylib 190 . . . . . . 7  |-  ( ph  ->  ( X  \  ( X  \  A ) )  =  A )
1413adantr 453 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  ( X  \  A
) )  =  A )
15 conclo.1 . . . . . . . . . 10  |-  ( ph  ->  J  e.  Con )
1615adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  J  e.  Con )
17 conndisj.4 . . . . . . . . . 10  |-  ( ph  ->  B  e.  J )
1817adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  e.  J )
19 conndisj.5 . . . . . . . . . 10  |-  ( ph  ->  B  =/=  (/) )
2019adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  =/=  (/) )
215iscon 17477 . . . . . . . . . . . . . 14  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
2221simplbi 448 . . . . . . . . . . . . 13  |-  ( J  e.  Con  ->  J  e.  Top )
2315, 22syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  Top )
245opncld 17098 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  ( X  \  A
)  e.  ( Clsd `  J ) )
2523, 2, 24syl2anc 644 . . . . . . . . . . 11  |-  ( ph  ->  ( X  \  A
)  e.  ( Clsd `  J ) )
2625adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  A )  e.  (
Clsd `  J )
)
2710, 26eqeltrrd 2512 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  e.  ( Clsd `  J )
)
285, 16, 18, 20, 27conclo 17479 . . . . . . . 8  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  =  X )
2928difeq2d 3466 . . . . . . 7  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  B )  =  ( X  \  X ) )
30 difid 3697 . . . . . . 7  |-  ( X 
\  X )  =  (/)
3129, 30syl6eq 2485 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  B )  =  (/) )
3211, 14, 313eqtr3d 2477 . . . . 5  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  A  =  (/) )
3332ex 425 . . . 4  |-  ( ph  ->  ( ( X  \  A )  =  B  ->  A  =  (/) ) )
349, 33sylbid 208 . . 3  |-  ( ph  ->  ( ( A  u.  B )  =  X  ->  A  =  (/) ) )
3534necon3d 2640 . 2  |-  ( ph  ->  ( A  =/=  (/)  ->  ( A  u.  B )  =/=  X ) )
361, 35mpd 15 1  |-  ( ph  ->  ( A  u.  B
)  =/=  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600    \ cdif 3318    u. cun 3319    i^i cin 3320    C_ wss 3321   (/)c0 3629   {cpr 3816   U.cuni 4016   ` cfv 5455   Topctop 16959   Clsdccld 17081   Conccon 17475
This theorem is referenced by:  dfcon2  17483
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-top 16964  df-cld 17084  df-con 17476
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