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Theorem conndisj 19032
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 4133 . . . . . . 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 3415 . . . . 5  |-  ( ph  ->  A  C_  X )
7 conndisj.6 . . . . 5  |-  ( ph  ->  ( A  i^i  B
)  =  (/) )
8 uneqdifeq 3779 . . . . 5  |-  ( ( A  C_  X  /\  ( A  i^i  B )  =  (/) )  ->  (
( A  u.  B
)  =  X  <->  ( X  \  A )  =  B ) )
96, 7, 8syl2anc 661 . . . 4  |-  ( ph  ->  ( ( A  u.  B )  =  X  <-> 
( X  \  A
)  =  B ) )
10 simpr 461 . . . . . . 7  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  A )  =  B )
1110difeq2d 3486 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  ( X  \  A
) )  =  ( X  \  B ) )
12 dfss4 3596 . . . . . . . 8  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
136, 12sylib 196 . . . . . . 7  |-  ( ph  ->  ( X  \  ( X  \  A ) )  =  A )
1413adantr 465 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  ( X  \  A
) )  =  A )
15 conclo.1 . . . . . . . . . 10  |-  ( ph  ->  J  e.  Con )
1615adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  J  e.  Con )
17 conndisj.4 . . . . . . . . . 10  |-  ( ph  ->  B  e.  J )
1817adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  e.  J )
19 conndisj.5 . . . . . . . . . 10  |-  ( ph  ->  B  =/=  (/) )
2019adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  =/=  (/) )
215iscon 19029 . . . . . . . . . . . . . 14  |-  ( J  e.  Con  <->  ( J  e.  Top  /\  ( J  i^i  ( Clsd `  J
) )  =  { (/)
,  X } ) )
2221simplbi 460 . . . . . . . . . . . . 13  |-  ( J  e.  Con  ->  J  e.  Top )
2315, 22syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  Top )
245opncld 18649 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  A  e.  J )  ->  ( X  \  A
)  e.  ( Clsd `  J ) )
2523, 2, 24syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( X  \  A
)  e.  ( Clsd `  J ) )
2625adantr 465 . . . . . . . . . 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 19031 . . . . . . . 8  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  B  =  X )
2928difeq2d 3486 . . . . . . 7  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  B )  =  ( X  \  X ) )
30 difid 3759 . . . . . . 7  |-  ( X 
\  X )  =  (/)
3129, 30syl6eq 2491 . . . . . 6  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  ( X  \  B )  =  (/) )
3211, 14, 313eqtr3d 2483 . . . . 5  |-  ( (
ph  /\  ( X  \  A )  =  B )  ->  A  =  (/) )
3332ex 434 . . . 4  |-  ( ph  ->  ( ( X  \  A )  =  B  ->  A  =  (/) ) )
349, 33sylbid 215 . . 3  |-  ( ph  ->  ( ( A  u.  B )  =  X  ->  A  =  (/) ) )
3534necon3d 2658 . 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 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618    \ cdif 3337    u. cun 3338    i^i cin 3339    C_ wss 3340   (/)c0 3649   {cpr 3891   U.cuni 4103   ` cfv 5430   Topctop 18510   Clsdccld 18632   Conccon 19027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-top 18515  df-cld 18635  df-con 19028
This theorem is referenced by:  dfcon2  19035
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