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Theorem indiscld 17157
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 17068 . . . . 5  |-  { (/) ,  A }  e.  Top
2 indisuni 17069 . . . . . 6  |-  (  _I 
`  A )  = 
U. { (/) ,  A }
32iscld 17093 . . . . 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 445 . . . . . 6  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  x  C_  (  _I  `  A ) )
6 dfss4 3577 . . . . . 6  |-  ( x 
C_  (  _I  `  A )  <->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  x )
75, 6sylib 190 . . . . 5  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  x )
8 simpr 449 . . . . . . 7  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \  x )  e.  { (/)
,  A } )
9 indislem 17066 . . . . . . 7  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
108, 9syl6eleqr 2529 . . . . . 6  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \  x )  e.  { (/)
,  (  _I  `  A ) } )
11 elpri 3836 . . . . . 6  |-  ( ( (  _I  `  A
)  \  x )  e.  { (/) ,  (  _I 
`  A ) }  ->  ( ( (  _I  `  A ) 
\  x )  =  (/)  \/  ( (  _I 
`  A )  \  x )  =  (  _I  `  A ) ) )
12 difeq2 3461 . . . . . . . . 9  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  ( (  _I  `  A
)  \  (/) ) )
13 dif0 3700 . . . . . . . . 9  |-  ( (  _I  `  A ) 
\  (/) )  =  (  _I  `  A )
1412, 13syl6eq 2486 . . . . . . . 8  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  (  _I  `  A ) )
15 fvex 5744 . . . . . . . . . 10  |-  (  _I 
`  A )  e. 
_V
1615prid2 3915 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
{ (/) ,  (  _I 
`  A ) }
1716, 9eleqtri 2510 . . . . . . . 8  |-  (  _I 
`  A )  e. 
{ (/) ,  A }
1814, 17syl6eqel 2526 . . . . . . 7  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
19 difeq2 3461 . . . . . . . . 9  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  ( (  _I  `  A
)  \  (  _I  `  A ) ) )
20 difid 3698 . . . . . . . . 9  |-  ( (  _I  `  A ) 
\  (  _I  `  A ) )  =  (/)
2119, 20syl6eq 2486 . . . . . . . 8  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  (/) )
22 0ex 4341 . . . . . . . . 9  |-  (/)  e.  _V
2322prid1 3914 . . . . . . . 8  |-  (/)  e.  { (/)
,  A }
2421, 23syl6eqel 2526 . . . . . . 7  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
2518, 24jaoi 370 . . . . . 6  |-  ( ( ( (  _I  `  A )  \  x
)  =  (/)  \/  (
(  _I  `  A
)  \  x )  =  (  _I  `  A
) )  ->  (
(  _I  `  A
)  \  ( (  _I  `  A )  \  x ) )  e. 
{ (/) ,  A }
)
2610, 11, 253syl 19 . . . . 5  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
277, 26eqeltrrd 2513 . . . 4  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  x  e.  {
(/) ,  A }
)
284, 27sylbi 189 . . 3  |-  ( x  e.  ( Clsd `  { (/)
,  A } )  ->  x  e.  { (/)
,  A } )
2928ssriv 3354 . 2  |-  ( Clsd `  { (/) ,  A }
)  C_  { (/) ,  A }
30 0cld 17104 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  (/)  e.  (
Clsd `  { (/) ,  A } ) )
311, 30ax-mp 8 . . . 4  |-  (/)  e.  (
Clsd `  { (/) ,  A } )
322topcld 17101 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  (  _I  `  A )  e.  (
Clsd `  { (/) ,  A } ) )
331, 32ax-mp 8 . . . 4  |-  (  _I 
`  A )  e.  ( Clsd `  { (/)
,  A } )
34 prssi 3956 . . . 4  |-  ( (
(/)  e.  ( Clsd `  { (/) ,  A }
)  /\  (  _I  `  A )  e.  (
Clsd `  { (/) ,  A } ) )  ->  { (/) ,  (  _I 
`  A ) } 
C_  ( Clsd `  { (/)
,  A } ) )
3531, 33, 34mp2an 655 . . 3  |-  { (/) ,  (  _I  `  A
) }  C_  ( Clsd `  { (/) ,  A } )
369, 35eqsstr3i 3381 . 2  |-  { (/) ,  A }  C_  ( Clsd `  { (/) ,  A } )
3729, 36eqssi 3366 1  |-  ( Clsd `  { (/) ,  A }
)  =  { (/) ,  A }
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3319    C_ wss 3322   (/)c0 3630   {cpr 3817    _I cid 4495   ` cfv 5456   Topctop 16960   Clsdccld 17082
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-top 16965  df-topon 16968  df-cld 17085
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