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Theorem indiscld 20156
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 20066 . . . . 5  |-  { (/) ,  A }  e.  Top
2 indisuni 20067 . . . . . 6  |-  (  _I 
`  A )  = 
U. { (/) ,  A }
32iscld 20091 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  ( x  e.  ( Clsd `  { (/)
,  A } )  <-> 
( x  C_  (  _I  `  A )  /\  ( (  _I  `  A )  \  x
)  e.  { (/) ,  A } ) ) )
41, 3ax-mp 5 . . . 4  |-  ( x  e.  ( Clsd `  { (/)
,  A } )  <-> 
( x  C_  (  _I  `  A )  /\  ( (  _I  `  A )  \  x
)  e.  { (/) ,  A } ) )
5 simpl 463 . . . . . 6  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  x  C_  (  _I  `  A ) )
6 dfss4 3689 . . . . . 6  |-  ( x 
C_  (  _I  `  A )  <->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  x )
75, 6sylib 201 . . . . 5  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  x )
8 simpr 467 . . . . . . 7  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \  x )  e.  { (/)
,  A } )
9 indislem 20064 . . . . . . 7  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
108, 9syl6eleqr 2551 . . . . . 6  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \  x )  e.  { (/)
,  (  _I  `  A ) } )
11 elpri 3997 . . . . . 6  |-  ( ( (  _I  `  A
)  \  x )  e.  { (/) ,  (  _I 
`  A ) }  ->  ( ( (  _I  `  A ) 
\  x )  =  (/)  \/  ( (  _I 
`  A )  \  x )  =  (  _I  `  A ) ) )
12 difeq2 3557 . . . . . . . . 9  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  ( (  _I  `  A
)  \  (/) ) )
13 dif0 3849 . . . . . . . . 9  |-  ( (  _I  `  A ) 
\  (/) )  =  (  _I  `  A )
1412, 13syl6eq 2512 . . . . . . . 8  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  (  _I  `  A ) )
15 fvex 5898 . . . . . . . . . 10  |-  (  _I 
`  A )  e. 
_V
1615prid2 4094 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
{ (/) ,  (  _I 
`  A ) }
1716, 9eleqtri 2538 . . . . . . . 8  |-  (  _I 
`  A )  e. 
{ (/) ,  A }
1814, 17syl6eqel 2548 . . . . . . 7  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
19 difeq2 3557 . . . . . . . . 9  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  ( (  _I  `  A
)  \  (  _I  `  A ) ) )
20 difid 3847 . . . . . . . . 9  |-  ( (  _I  `  A ) 
\  (  _I  `  A ) )  =  (/)
2119, 20syl6eq 2512 . . . . . . . 8  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  (/) )
22 0ex 4549 . . . . . . . . 9  |-  (/)  e.  _V
2322prid1 4093 . . . . . . . 8  |-  (/)  e.  { (/)
,  A }
2421, 23syl6eqel 2548 . . . . . . 7  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
2518, 24jaoi 385 . . . . . 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 2541 . . . 4  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  x  e.  {
(/) ,  A }
)
284, 27sylbi 200 . . 3  |-  ( x  e.  ( Clsd `  { (/)
,  A } )  ->  x  e.  { (/)
,  A } )
2928ssriv 3448 . 2  |-  ( Clsd `  { (/) ,  A }
)  C_  { (/) ,  A }
30 0cld 20102 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  (/)  e.  (
Clsd `  { (/) ,  A } ) )
311, 30ax-mp 5 . . . 4  |-  (/)  e.  (
Clsd `  { (/) ,  A } )
322topcld 20099 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  (  _I  `  A )  e.  (
Clsd `  { (/) ,  A } ) )
331, 32ax-mp 5 . . . 4  |-  (  _I 
`  A )  e.  ( Clsd `  { (/)
,  A } )
34 prssi 4141 . . . 4  |-  ( (
(/)  e.  ( Clsd `  { (/) ,  A }
)  /\  (  _I  `  A )  e.  (
Clsd `  { (/) ,  A } ) )  ->  { (/) ,  (  _I 
`  A ) } 
C_  ( Clsd `  { (/)
,  A } ) )
3531, 33, 34mp2an 683 . . 3  |-  { (/) ,  (  _I  `  A
) }  C_  ( Clsd `  { (/) ,  A } )
369, 35eqsstr3i 3475 . 2  |-  { (/) ,  A }  C_  ( Clsd `  { (/) ,  A } )
3729, 36eqssi 3460 1  |-  ( Clsd `  { (/) ,  A }
)  =  { (/) ,  A }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    \/ wo 374    /\ wa 375    = wceq 1455    e. wcel 1898    \ cdif 3413    C_ wss 3416   (/)c0 3743   {cpr 3982    _I cid 4763   ` cfv 5601   Topctop 19966   Clsdccld 20080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-top 19970  df-topon 19972  df-cld 20083
This theorem is referenced by: (None)
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