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Theorem indiscld 19678
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 19588 . . . . 5  |-  { (/) ,  A }  e.  Top
2 indisuni 19589 . . . . . 6  |-  (  _I 
`  A )  = 
U. { (/) ,  A }
32iscld 19613 . . . . 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 455 . . . . . 6  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  x  C_  (  _I  `  A ) )
6 dfss4 3657 . . . . . 6  |-  ( x 
C_  (  _I  `  A )  <->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  x )
75, 6sylib 196 . . . . 5  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  x )
8 simpr 459 . . . . . . 7  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \  x )  e.  { (/)
,  A } )
9 indislem 19586 . . . . . . 7  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
108, 9syl6eleqr 2481 . . . . . 6  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \  x )  e.  { (/)
,  (  _I  `  A ) } )
11 elpri 3964 . . . . . 6  |-  ( ( (  _I  `  A
)  \  x )  e.  { (/) ,  (  _I 
`  A ) }  ->  ( ( (  _I  `  A ) 
\  x )  =  (/)  \/  ( (  _I 
`  A )  \  x )  =  (  _I  `  A ) ) )
12 difeq2 3530 . . . . . . . . 9  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  ( (  _I  `  A
)  \  (/) ) )
13 dif0 3814 . . . . . . . . 9  |-  ( (  _I  `  A ) 
\  (/) )  =  (  _I  `  A )
1412, 13syl6eq 2439 . . . . . . . 8  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  (  _I  `  A ) )
15 fvex 5784 . . . . . . . . . 10  |-  (  _I 
`  A )  e. 
_V
1615prid2 4053 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
{ (/) ,  (  _I 
`  A ) }
1716, 9eleqtri 2468 . . . . . . . 8  |-  (  _I 
`  A )  e. 
{ (/) ,  A }
1814, 17syl6eqel 2478 . . . . . . 7  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
19 difeq2 3530 . . . . . . . . 9  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  ( (  _I  `  A
)  \  (  _I  `  A ) ) )
20 difid 3812 . . . . . . . . 9  |-  ( (  _I  `  A ) 
\  (  _I  `  A ) )  =  (/)
2119, 20syl6eq 2439 . . . . . . . 8  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  (/) )
22 0ex 4497 . . . . . . . . 9  |-  (/)  e.  _V
2322prid1 4052 . . . . . . . 8  |-  (/)  e.  { (/)
,  A }
2421, 23syl6eqel 2478 . . . . . . 7  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
2518, 24jaoi 377 . . . . . 6  |-  ( ( ( (  _I  `  A )  \  x
)  =  (/)  \/  (
(  _I  `  A
)  \  x )  =  (  _I  `  A
) )  ->  (
(  _I  `  A
)  \  ( (  _I  `  A )  \  x ) )  e. 
{ (/) ,  A }
)
2610, 11, 253syl 20 . . . . 5  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
277, 26eqeltrrd 2471 . . . 4  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  x  e.  {
(/) ,  A }
)
284, 27sylbi 195 . . 3  |-  ( x  e.  ( Clsd `  { (/)
,  A } )  ->  x  e.  { (/)
,  A } )
2928ssriv 3421 . 2  |-  ( Clsd `  { (/) ,  A }
)  C_  { (/) ,  A }
30 0cld 19624 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  (/)  e.  (
Clsd `  { (/) ,  A } ) )
311, 30ax-mp 5 . . . 4  |-  (/)  e.  (
Clsd `  { (/) ,  A } )
322topcld 19621 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  (  _I  `  A )  e.  (
Clsd `  { (/) ,  A } ) )
331, 32ax-mp 5 . . . 4  |-  (  _I 
`  A )  e.  ( Clsd `  { (/)
,  A } )
34 prssi 4100 . . . 4  |-  ( (
(/)  e.  ( Clsd `  { (/) ,  A }
)  /\  (  _I  `  A )  e.  (
Clsd `  { (/) ,  A } ) )  ->  { (/) ,  (  _I 
`  A ) } 
C_  ( Clsd `  { (/)
,  A } ) )
3531, 33, 34mp2an 670 . . 3  |-  { (/) ,  (  _I  `  A
) }  C_  ( Clsd `  { (/) ,  A } )
369, 35eqsstr3i 3448 . 2  |-  { (/) ,  A }  C_  ( Clsd `  { (/) ,  A } )
3729, 36eqssi 3433 1  |-  ( Clsd `  { (/) ,  A }
)  =  { (/) ,  A }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1826    \ cdif 3386    C_ wss 3389   (/)c0 3711   {cpr 3946    _I cid 4704   ` cfv 5496   Topctop 19479   Clsdccld 19602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-top 19484  df-topon 19487  df-cld 19605
This theorem is referenced by: (None)
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