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Theorem distop 17052
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 4028 . . . . . 6  |-  ( x 
C_  ~P A  ->  U. x  C_ 
U. ~P A )
2 unipw 4406 . . . . . 6  |-  U. ~P A  =  A
31, 2syl6sseq 3386 . . . . 5  |-  ( x 
C_  ~P A  ->  U. x  C_  A )
4 vex 2951 . . . . . . 7  |-  x  e. 
_V
54uniex 4697 . . . . . 6  |-  U. x  e.  _V
65elpw 3797 . . . . 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 3797 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
11 vex 2951 . . . . . . . . 9  |-  y  e. 
_V
1211elpw 3797 . . . . . . . 8  |-  ( y  e.  ~P A  <->  y  C_  A )
13 ssinss1 3561 . . . . . . . . . 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 4337 . . . . . . . . . 10  |-  ( x  i^i  y )  e. 
_V
1615elpw 3797 . . . . . . . . 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 2780 . . . 4  |-  ( x  e.  ~P A  ->  A. y  e.  ~P  A ( x  i^i  y )  e.  ~P A )
2221rgen 2763 . . 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 4375 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
25 istopg 16960 . . 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 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   Topctop 16950
This theorem is referenced by:  distopon  17053  distps  17071  discld  17145  restdis  17234  dishaus  17438  discmp  17453  dis2ndc  17515  dislly  17552  dis1stc  17554  txdis  17656  xkopt  17679  xkofvcn  17708  symgtgp  18123  locfindis  26376
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-pw 3793  df-sn 3812  df-pr 3813  df-uni 4008  df-top 16955
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