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Theorem distop 19789
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 4212 . . . . . 6  |-  ( x 
C_  ~P A  ->  U. x  C_ 
U. ~P A )
2 unipw 4641 . . . . . 6  |-  U. ~P A  =  A
31, 2syl6sseq 3488 . . . . 5  |-  ( x 
C_  ~P A  ->  U. x  C_  A )
4 vex 3062 . . . . . . 7  |-  x  e. 
_V
54uniex 6578 . . . . . 6  |-  U. x  e.  _V
65elpw 3961 . . . . 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 1639 . . 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 3962 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
11 selpw 3962 . . . . . . . 8  |-  ( y  e.  ~P A  <->  y  C_  A )
12 ssinss1 3667 . . . . . . . . . 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 3062 . . . . . . . . . . 11  |-  y  e. 
_V
1514inex2 4536 . . . . . . . . . 10  |-  ( x  i^i  y )  e. 
_V
1615elpw 3961 . . . . . . . . 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 29 . . . . . 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 2816 . . . 4  |-  ( x  e.  ~P A  ->  A. y  e.  ~P  A ( x  i^i  y )  e.  ~P A )
2221rgen 2764 . . 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 4578 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
25 istopg 19696 . . 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 17 . 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 923 1  |-  ( A  e.  V  ->  ~P A  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403    e. wcel 1842   A.wral 2754   _Vcvv 3059    i^i cin 3413    C_ wss 3414   ~Pcpw 3955   U.cuni 4191   Topctop 19686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-pw 3957  df-sn 3973  df-pr 3975  df-uni 4192  df-top 19691
This theorem is referenced by:  distopon  19790  distps  19808  discld  19883  restdis  19972  dishaus  20176  discmp  20191  dis2ndc  20253  dislly  20290  dis1stc  20292  dissnlocfin  20322  locfindis  20323  txdis  20425  xkopt  20448  xkofvcn  20477  symgtgp  20892  dispcmp  28315
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