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Theorem indiscld 16828
Description: The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
indiscld  |-  ( Clsd `  { (/) ,  A }
)  =  { (/) ,  A }

Proof of Theorem indiscld
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 indistop 16739 . . . . 5  |-  { (/) ,  A }  e.  Top
2 indisuni 16740 . . . . . 6  |-  (  _I 
`  A )  = 
U. { (/) ,  A }
32iscld 16764 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  ( x  e.  ( Clsd `  { (/)
,  A } )  <-> 
( x  C_  (  _I  `  A )  /\  ( (  _I  `  A )  \  x
)  e.  { (/) ,  A } ) ) )
41, 3ax-mp 8 . . . 4  |-  ( x  e.  ( Clsd `  { (/)
,  A } )  <-> 
( x  C_  (  _I  `  A )  /\  ( (  _I  `  A )  \  x
)  e.  { (/) ,  A } ) )
5 simpl 443 . . . . . 6  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  x  C_  (  _I  `  A ) )
6 dfss4 3403 . . . . . 6  |-  ( x 
C_  (  _I  `  A )  <->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  x )
75, 6sylib 188 . . . . 5  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  x )
8 simpr 447 . . . . . . 7  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \  x )  e.  { (/)
,  A } )
9 indislem 16737 . . . . . . 7  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
108, 9syl6eleqr 2374 . . . . . 6  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \  x )  e.  { (/)
,  (  _I  `  A ) } )
11 elpri 3660 . . . . . 6  |-  ( ( (  _I  `  A
)  \  x )  e.  { (/) ,  (  _I 
`  A ) }  ->  ( ( (  _I  `  A ) 
\  x )  =  (/)  \/  ( (  _I 
`  A )  \  x )  =  (  _I  `  A ) ) )
12 difeq2 3288 . . . . . . . . 9  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  ( (  _I  `  A
)  \  (/) ) )
13 dif0 3524 . . . . . . . . 9  |-  ( (  _I  `  A ) 
\  (/) )  =  (  _I  `  A )
1412, 13syl6eq 2331 . . . . . . . 8  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  (  _I  `  A ) )
15 fvex 5539 . . . . . . . . . 10  |-  (  _I 
`  A )  e. 
_V
1615prid2 3735 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
{ (/) ,  (  _I 
`  A ) }
1716, 9eleqtri 2355 . . . . . . . 8  |-  (  _I 
`  A )  e. 
{ (/) ,  A }
1814, 17syl6eqel 2371 . . . . . . 7  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
19 difeq2 3288 . . . . . . . . 9  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  ( (  _I  `  A
)  \  (  _I  `  A ) ) )
20 difid 3522 . . . . . . . . 9  |-  ( (  _I  `  A ) 
\  (  _I  `  A ) )  =  (/)
2119, 20syl6eq 2331 . . . . . . . 8  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  (/) )
22 0ex 4150 . . . . . . . . 9  |-  (/)  e.  _V
2322prid1 3734 . . . . . . . 8  |-  (/)  e.  { (/)
,  A }
2421, 23syl6eqel 2371 . . . . . . 7  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
2518, 24jaoi 368 . . . . . 6  |-  ( ( ( (  _I  `  A )  \  x
)  =  (/)  \/  (
(  _I  `  A
)  \  x )  =  (  _I  `  A
) )  ->  (
(  _I  `  A
)  \  ( (  _I  `  A )  \  x ) )  e. 
{ (/) ,  A }
)
2610, 11, 253syl 18 . . . . 5  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
277, 26eqeltrrd 2358 . . . 4  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  x  e.  {
(/) ,  A }
)
284, 27sylbi 187 . . 3  |-  ( x  e.  ( Clsd `  { (/)
,  A } )  ->  x  e.  { (/)
,  A } )
2928ssriv 3184 . 2  |-  ( Clsd `  { (/) ,  A }
)  C_  { (/) ,  A }
30 0cld 16775 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  (/)  e.  (
Clsd `  { (/) ,  A } ) )
311, 30ax-mp 8 . . . 4  |-  (/)  e.  (
Clsd `  { (/) ,  A } )
322topcld 16772 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  (  _I  `  A )  e.  (
Clsd `  { (/) ,  A } ) )
331, 32ax-mp 8 . . . 4  |-  (  _I 
`  A )  e.  ( Clsd `  { (/)
,  A } )
34 prssi 3771 . . . 4  |-  ( (
(/)  e.  ( Clsd `  { (/) ,  A }
)  /\  (  _I  `  A )  e.  (
Clsd `  { (/) ,  A } ) )  ->  { (/) ,  (  _I 
`  A ) } 
C_  ( Clsd `  { (/)
,  A } ) )
3531, 33, 34mp2an 653 . . 3  |-  { (/) ,  (  _I  `  A
) }  C_  ( Clsd `  { (/) ,  A } )
369, 35eqsstr3i 3209 . 2  |-  { (/) ,  A }  C_  ( Clsd `  { (/) ,  A } )
3729, 36eqssi 3195 1  |-  ( Clsd `  { (/) ,  A }
)  =  { (/) ,  A }
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   (/)c0 3455   {cpr 3641    _I cid 4304   ` cfv 5255   Topctop 16631   Clsdccld 16753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-top 16636  df-topon 16639  df-cld 16756
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