MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  indiscld Structured version   Unicode version

Theorem indiscld 19355
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 19266 . . . . 5  |-  { (/) ,  A }  e.  Top
2 indisuni 19267 . . . . . 6  |-  (  _I 
`  A )  = 
U. { (/) ,  A }
32iscld 19291 . . . . 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 457 . . . . . 6  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  x  C_  (  _I  `  A ) )
6 dfss4 3732 . . . . . 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 461 . . . . . . 7  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \  x )  e.  { (/)
,  A } )
9 indislem 19264 . . . . . . 7  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
108, 9syl6eleqr 2566 . . . . . 6  |-  ( ( x  C_  (  _I  `  A )  /\  (
(  _I  `  A
)  \  x )  e.  { (/) ,  A }
)  ->  ( (  _I  `  A )  \  x )  e.  { (/)
,  (  _I  `  A ) } )
11 elpri 4047 . . . . . 6  |-  ( ( (  _I  `  A
)  \  x )  e.  { (/) ,  (  _I 
`  A ) }  ->  ( ( (  _I  `  A ) 
\  x )  =  (/)  \/  ( (  _I 
`  A )  \  x )  =  (  _I  `  A ) ) )
12 difeq2 3616 . . . . . . . . 9  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  ( (  _I  `  A
)  \  (/) ) )
13 dif0 3897 . . . . . . . . 9  |-  ( (  _I  `  A ) 
\  (/) )  =  (  _I  `  A )
1412, 13syl6eq 2524 . . . . . . . 8  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  (  _I  `  A ) )
15 fvex 5874 . . . . . . . . . 10  |-  (  _I 
`  A )  e. 
_V
1615prid2 4136 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
{ (/) ,  (  _I 
`  A ) }
1716, 9eleqtri 2553 . . . . . . . 8  |-  (  _I 
`  A )  e. 
{ (/) ,  A }
1814, 17syl6eqel 2563 . . . . . . 7  |-  ( ( (  _I  `  A
)  \  x )  =  (/)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
19 difeq2 3616 . . . . . . . . 9  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  ( (  _I  `  A
)  \  (  _I  `  A ) ) )
20 difid 3895 . . . . . . . . 9  |-  ( (  _I  `  A ) 
\  (  _I  `  A ) )  =  (/)
2119, 20syl6eq 2524 . . . . . . . 8  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  =  (/) )
22 0ex 4577 . . . . . . . . 9  |-  (/)  e.  _V
2322prid1 4135 . . . . . . . 8  |-  (/)  e.  { (/)
,  A }
2421, 23syl6eqel 2563 . . . . . . 7  |-  ( ( (  _I  `  A
)  \  x )  =  (  _I  `  A
)  ->  ( (  _I  `  A )  \ 
( (  _I  `  A )  \  x
) )  e.  { (/)
,  A } )
2518, 24jaoi 379 . . . . . 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 2556 . . . 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 3508 . 2  |-  ( Clsd `  { (/) ,  A }
)  C_  { (/) ,  A }
30 0cld 19302 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  (/)  e.  (
Clsd `  { (/) ,  A } ) )
311, 30ax-mp 5 . . . 4  |-  (/)  e.  (
Clsd `  { (/) ,  A } )
322topcld 19299 . . . . 5  |-  ( {
(/) ,  A }  e.  Top  ->  (  _I  `  A )  e.  (
Clsd `  { (/) ,  A } ) )
331, 32ax-mp 5 . . . 4  |-  (  _I 
`  A )  e.  ( Clsd `  { (/)
,  A } )
34 prssi 4183 . . . 4  |-  ( (
(/)  e.  ( Clsd `  { (/) ,  A }
)  /\  (  _I  `  A )  e.  (
Clsd `  { (/) ,  A } ) )  ->  { (/) ,  (  _I 
`  A ) } 
C_  ( Clsd `  { (/)
,  A } ) )
3531, 33, 34mp2an 672 . . 3  |-  { (/) ,  (  _I  `  A
) }  C_  ( Clsd `  { (/) ,  A } )
369, 35eqsstr3i 3535 . 2  |-  { (/) ,  A }  C_  ( Clsd `  { (/) ,  A } )
3729, 36eqssi 3520 1  |-  ( Clsd `  { (/) ,  A }
)  =  { (/) ,  A }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3473    C_ wss 3476   (/)c0 3785   {cpr 4029    _I cid 4790   ` cfv 5586   Topctop 19158   Clsdccld 19280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-top 19163  df-topon 19166  df-cld 19283
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator