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Theorem distop 18600
Description: The discrete topology on a set  A. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
distop  |-  ( A  e.  V  ->  ~P A  e.  Top )

Proof of Theorem distop
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniss 4112 . . . . . 6  |-  ( x 
C_  ~P A  ->  U. x  C_ 
U. ~P A )
2 unipw 4542 . . . . . 6  |-  U. ~P A  =  A
31, 2syl6sseq 3402 . . . . 5  |-  ( x 
C_  ~P A  ->  U. x  C_  A )
4 vex 2975 . . . . . . 7  |-  x  e. 
_V
54uniex 6376 . . . . . 6  |-  U. x  e.  _V
65elpw 3866 . . . . 5  |-  ( U. x  e.  ~P A  <->  U. x  C_  A )
73, 6sylibr 212 . . . 4  |-  ( x 
C_  ~P A  ->  U. x  e.  ~P A )
87ax-gen 1591 . . 3  |-  A. x
( x  C_  ~P A  ->  U. x  e.  ~P A )
98a1i 11 . 2  |-  ( A  e.  V  ->  A. x
( x  C_  ~P A  ->  U. x  e.  ~P A ) )
10 selpw 3867 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
11 selpw 3867 . . . . . . . 8  |-  ( y  e.  ~P A  <->  y  C_  A )
12 ssinss1 3578 . . . . . . . . . 10  |-  ( x 
C_  A  ->  (
x  i^i  y )  C_  A )
1312a1i 11 . . . . . . . . 9  |-  ( y 
C_  A  ->  (
x  C_  A  ->  ( x  i^i  y ) 
C_  A ) )
14 vex 2975 . . . . . . . . . . 11  |-  y  e. 
_V
1514inex2 4434 . . . . . . . . . 10  |-  ( x  i^i  y )  e. 
_V
1615elpw 3866 . . . . . . . . 9  |-  ( ( x  i^i  y )  e.  ~P A  <->  ( x  i^i  y )  C_  A
)
1713, 16syl6ibr 227 . . . . . . . 8  |-  ( y 
C_  A  ->  (
x  C_  A  ->  ( x  i^i  y )  e.  ~P A ) )
1811, 17sylbi 195 . . . . . . 7  |-  ( y  e.  ~P A  -> 
( x  C_  A  ->  ( x  i^i  y
)  e.  ~P A
) )
1918com12 31 . . . . . 6  |-  ( x 
C_  A  ->  (
y  e.  ~P A  ->  ( x  i^i  y
)  e.  ~P A
) )
2010, 19sylbi 195 . . . . 5  |-  ( x  e.  ~P A  -> 
( y  e.  ~P A  ->  ( x  i^i  y )  e.  ~P A ) )
2120ralrimiv 2798 . . . 4  |-  ( x  e.  ~P A  ->  A. y  e.  ~P  A ( x  i^i  y )  e.  ~P A )
2221rgen 2781 . . 3  |-  A. x  e.  ~P  A A. y  e.  ~P  A ( x  i^i  y )  e. 
~P A
2322a1i 11 . 2  |-  ( A  e.  V  ->  A. x  e.  ~P  A A. y  e.  ~P  A ( x  i^i  y )  e. 
~P A )
24 pwexg 4476 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
25 istopg 18508 . . 3  |-  ( ~P A  e.  _V  ->  ( ~P A  e.  Top  <->  ( A. x ( x  C_  ~P A  ->  U. x  e.  ~P A )  /\  A. x  e.  ~P  A A. y  e.  ~P  A ( x  i^i  y )  e.  ~P A ) ) )
2624, 25syl 16 . 2  |-  ( A  e.  V  ->  ( ~P A  e.  Top  <->  ( A. x ( x  C_  ~P A  ->  U. x  e.  ~P A )  /\  A. x  e.  ~P  A A. y  e.  ~P  A ( x  i^i  y )  e.  ~P A ) ) )
279, 23, 26mpbir2and 913 1  |-  ( A  e.  V  ->  ~P A  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    e. wcel 1756   A.wral 2715   _Vcvv 2972    i^i cin 3327    C_ wss 3328   ~Pcpw 3860   U.cuni 4091   Topctop 18498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-pw 3862  df-sn 3878  df-pr 3880  df-uni 4092  df-top 18503
This theorem is referenced by:  distopon  18601  distps  18619  discld  18693  restdis  18782  dishaus  18986  discmp  19001  dis2ndc  19064  dislly  19101  dis1stc  19103  txdis  19205  xkopt  19228  xkofvcn  19257  symgtgp  19672  locfindis  28577
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