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Theorem conndisj 20362
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
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 4251 . . . . . . 7  |-  ( A  e.  J  ->  A  C_ 
U. J )
42, 3syl 17 . . . . . 6  |-  ( ph  ->  A  C_  U. J )
5 iscon.1 . . . . . 6  |-  X  = 
U. J
64, 5syl6sseqr 3517 . . . . 5  |-  ( ph  ->  A  C_  X )
7 conndisj.6 . . . . 5  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
8 uneqdifeq 3890 . . . . 5  |-  ( ( A  C_  X  /\  ( A  i^i  B )  =  (/) )  ->  (
( A  u.  B
)  =  X  <->  ( X  \  A )  =  B ) )
96, 7, 8syl2anc 665 . . . 4  |-  ( ph  ->  ( ( A  u.  B )  =  X  <-> 
( X  \  A
)  =  B ) )
10 simpr 462 . . . . . . 7  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  A )  =  B )
1110difeq2d 3589 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  ( X  \  A
) )  =  ( X  \  B ) )
12 dfss4 3713 . . . . . . . 8  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
136, 12sylib 199 . . . . . . 7  |-  ( ph  ->  ( X  \  ( X  \  A ) )  =  A )
1413adantr 466 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  ( X  \  A
) )  =  A )
15 conclo.1 . . . . . . . . . 10  |-  ( ph  ->  J  e.  Con )
1615adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  J  e.  Con )
17 conndisj.4 . . . . . . . . . 10  |-  ( ph  ->  B  e.  J )
1817adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  e.  J )
19 conndisj.5 . . . . . . . . . 10  |-  ( ph  ->  B  =/=  (/) )
2019adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  =/=  (/) )
215iscon 20359 . . . . . . . . . . . . . 14  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
2221simplbi 461 . . . . . . . . . . . . 13  |-  ( J  e.  Con  ->  J  e.  Top )
2315, 22syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  Top )
245opncld 19979 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  ( X  \  A
)  e.  ( Clsd `  J ) )
2523, 2, 24syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  ( X  \  A
)  e.  ( Clsd `  J ) )
2625adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  A )  e.  (
Clsd `  J )
)
2710, 26eqeltrrd 2518 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  e.  ( Clsd `  J )
)
285, 16, 18, 20, 27conclo 20361 . . . . . . . 8  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  =  X )
2928difeq2d 3589 . . . . . . 7  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  B )  =  ( X  \  X ) )
30 difid 3869 . . . . . . 7  |-  ( X 
\  X )  =  (/)
3129, 30syl6eq 2486 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  B )  =  (/) )
3211, 14, 313eqtr3d 2478 . . . . 5  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  A  =  (/) )
3332ex 435 . . . 4  |-  ( ph  ->  ( ( X  \  A )  =  B  ->  A  =  (/) ) )
349, 33sylbid 218 . . 3  |-  ( ph  ->  ( ( A  u.  B )  =  X  ->  A  =  (/) ) )
3534necon3d 2655 . 2  |-  ( ph  ->  ( A  =/=  (/)  ->  ( A  u.  B )  =/=  X ) )
361, 35mpd 15 1  |-  ( ph  ->  ( A  u.  B
)  =/=  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625    \ cdif 3439    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   {cpr 4004   U.cuni 4222   ` cfv 5601   Topctop 19848   Clsdccld 19962   Conccon 20357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-top 19852  df-cld 19965  df-con 20358
This theorem is referenced by:  dfcon2  20365
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