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Theorem distop 16749
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 3864 . . . . . 6  |-  ( x 
C_  ~P A  ->  U. x  C_ 
U. ~P A )
2 unipw 4240 . . . . . 6  |-  U. ~P A  =  A
31, 2syl6sseq 3237 . . . . 5  |-  ( x 
C_  ~P A  ->  U. x  C_  A )
4 vex 2804 . . . . . . 7  |-  x  e. 
_V
54uniex 4532 . . . . . 6  |-  U. x  e.  _V
65elpw 3644 . . . . 5  |-  ( U. x  e.  ~P A  <->  U. x  C_  A )
73, 6sylibr 203 . . . 4  |-  ( x 
C_  ~P A  ->  U. x  e.  ~P A )
87ax-gen 1536 . . 3  |-  A. x
( x  C_  ~P A  ->  U. x  e.  ~P A )
98a1i 10 . 2  |-  ( A  e.  V  ->  A. x
( x  C_  ~P A  ->  U. x  e.  ~P A ) )
104elpw 3644 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
11 vex 2804 . . . . . . . . 9  |-  y  e. 
_V
1211elpw 3644 . . . . . . . 8  |-  ( y  e.  ~P A  <->  y  C_  A )
13 ssinss1 3410 . . . . . . . . . 10  |-  ( x 
C_  A  ->  (
x  i^i  y )  C_  A )
1413a1i 10 . . . . . . . . 9  |-  ( y 
C_  A  ->  (
x  C_  A  ->  ( x  i^i  y ) 
C_  A ) )
1511inex2 4172 . . . . . . . . . 10  |-  ( x  i^i  y )  e. 
_V
1615elpw 3644 . . . . . . . . 9  |-  ( ( x  i^i  y )  e.  ~P A  <->  ( x  i^i  y )  C_  A
)
1714, 16syl6ibr 218 . . . . . . . 8  |-  ( y 
C_  A  ->  (
x  C_  A  ->  ( x  i^i  y )  e.  ~P A ) )
1812, 17sylbi 187 . . . . . . 7  |-  ( y  e.  ~P A  -> 
( x  C_  A  ->  ( x  i^i  y
)  e.  ~P A
) )
1918com12 27 . . . . . 6  |-  ( x 
C_  A  ->  (
y  e.  ~P A  ->  ( x  i^i  y
)  e.  ~P A
) )
2010, 19sylbi 187 . . . . 5  |-  ( x  e.  ~P A  -> 
( y  e.  ~P A  ->  ( x  i^i  y )  e.  ~P A ) )
2120ralrimiv 2638 . . . 4  |-  ( x  e.  ~P A  ->  A. y  e.  ~P  A ( x  i^i  y )  e.  ~P A )
2221rgen 2621 . . 3  |-  A. x  e.  ~P  A A. y  e.  ~P  A ( x  i^i  y )  e. 
~P A
2322a1i 10 . 2  |-  ( A  e.  V  ->  A. x  e.  ~P  A A. y  e.  ~P  A ( x  i^i  y )  e. 
~P A )
24 pwexg 4210 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
25 istopg 16657 . . 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 15 . 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 888 1  |-  ( A  e.  V  ->  ~P A  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    e. wcel 1696   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   Topctop 16647
This theorem is referenced by:  distopon  16750  distps  16768  discld  16842  restdis  16925  dishaus  17126  discmp  17141  dis2ndc  17202  dislly  17239  dis1stc  17241  txdis  17342  xkopt  17365  xkofvcn  17394  symgtgp  17800  usptoplem  25649  istopx  25650  locfindis  26408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660  df-uni 3844  df-top 16652
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