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Theorem distop 17060
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 4036 . . . . . 6  |-  ( x 
C_  ~P A  ->  U. x  C_ 
U. ~P A )
2 unipw 4414 . . . . . 6  |-  U. ~P A  =  A
31, 2syl6sseq 3394 . . . . 5  |-  ( x 
C_  ~P A  ->  U. x  C_  A )
4 vex 2959 . . . . . . 7  |-  x  e. 
_V
54uniex 4705 . . . . . 6  |-  U. x  e.  _V
65elpw 3805 . . . . 5  |-  ( U. x  e.  ~P A  <->  U. x  C_  A )
73, 6sylibr 204 . . . 4  |-  ( x 
C_  ~P A  ->  U. x  e.  ~P A )
87ax-gen 1555 . . 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 ) )
104elpw 3805 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
11 vex 2959 . . . . . . . . 9  |-  y  e. 
_V
1211elpw 3805 . . . . . . . 8  |-  ( y  e.  ~P A  <->  y  C_  A )
13 ssinss1 3569 . . . . . . . . . 10  |-  ( x 
C_  A  ->  (
x  i^i  y )  C_  A )
1413a1i 11 . . . . . . . . 9  |-  ( y 
C_  A  ->  (
x  C_  A  ->  ( x  i^i  y ) 
C_  A ) )
1511inex2 4345 . . . . . . . . . 10  |-  ( x  i^i  y )  e. 
_V
1615elpw 3805 . . . . . . . . 9  |-  ( ( x  i^i  y )  e.  ~P A  <->  ( x  i^i  y )  C_  A
)
1714, 16syl6ibr 219 . . . . . . . 8  |-  ( y 
C_  A  ->  (
x  C_  A  ->  ( x  i^i  y )  e.  ~P A ) )
1812, 17sylbi 188 . . . . . . 7  |-  ( y  e.  ~P A  -> 
( x  C_  A  ->  ( x  i^i  y
)  e.  ~P A
) )
1918com12 29 . . . . . 6  |-  ( x 
C_  A  ->  (
y  e.  ~P A  ->  ( x  i^i  y
)  e.  ~P A
) )
2010, 19sylbi 188 . . . . 5  |-  ( x  e.  ~P A  -> 
( y  e.  ~P A  ->  ( x  i^i  y )  e.  ~P A ) )
2120ralrimiv 2788 . . . 4  |-  ( x  e.  ~P A  ->  A. y  e.  ~P  A ( x  i^i  y )  e.  ~P A )
2221rgen 2771 . . 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 4383 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
25 istopg 16968 . . 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 889 1  |-  ( A  e.  V  ->  ~P A  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    e. wcel 1725   A.wral 2705   _Vcvv 2956    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   Topctop 16958
This theorem is referenced by:  distopon  17061  distps  17079  discld  17153  restdis  17242  dishaus  17446  discmp  17461  dis2ndc  17523  dislly  17560  dis1stc  17562  txdis  17664  xkopt  17687  xkofvcn  17716  symgtgp  18131  locfindis  26385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-pw 3801  df-sn 3820  df-pr 3821  df-uni 4016  df-top 16963
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