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Theorem distop 12243
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 3752 . . . . . 6  |-  ( x 
C_  ~P A  ->  U. x  C_ 
U. ~P A )
2 unipw 4134 . . . . . 6  |-  U. ~P A  =  A
31, 2sseqtrdi 3140 . . . . 5  |-  ( x 
C_  ~P A  ->  U. x  C_  A )
4 vuniex 4355 . . . . . 6  |-  U. x  e.  _V
54elpw 3511 . . . . 5  |-  ( U. x  e.  ~P A  <->  U. x  C_  A )
63, 5sylibr 133 . . . 4  |-  ( x 
C_  ~P A  ->  U. x  e.  ~P A )
76ax-gen 1425 . . 3  |-  A. x
( x  C_  ~P A  ->  U. x  e.  ~P A )
87a1i 9 . 2  |-  ( A  e.  V  ->  A. x
( x  C_  ~P A  ->  U. x  e.  ~P A ) )
9 velpw 3512 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
10 velpw 3512 . . . . . . . 8  |-  ( y  e.  ~P A  <->  y  C_  A )
11 ssinss1 3300 . . . . . . . . . 10  |-  ( x 
C_  A  ->  (
x  i^i  y )  C_  A )
1211a1i 9 . . . . . . . . 9  |-  ( y 
C_  A  ->  (
x  C_  A  ->  ( x  i^i  y ) 
C_  A ) )
13 vex 2684 . . . . . . . . . . 11  |-  y  e. 
_V
1413inex2 4058 . . . . . . . . . 10  |-  ( x  i^i  y )  e. 
_V
1514elpw 3511 . . . . . . . . 9  |-  ( ( x  i^i  y )  e.  ~P A  <->  ( x  i^i  y )  C_  A
)
1612, 15syl6ibr 161 . . . . . . . 8  |-  ( y 
C_  A  ->  (
x  C_  A  ->  ( x  i^i  y )  e.  ~P A ) )
1710, 16sylbi 120 . . . . . . 7  |-  ( y  e.  ~P A  -> 
( x  C_  A  ->  ( x  i^i  y
)  e.  ~P A
) )
1817com12 30 . . . . . 6  |-  ( x 
C_  A  ->  (
y  e.  ~P A  ->  ( x  i^i  y
)  e.  ~P A
) )
199, 18sylbi 120 . . . . 5  |-  ( x  e.  ~P A  -> 
( y  e.  ~P A  ->  ( x  i^i  y )  e.  ~P A ) )
2019ralrimiv 2502 . . . 4  |-  ( x  e.  ~P A  ->  A. y  e.  ~P  A ( x  i^i  y )  e.  ~P A )
2120rgen 2483 . . 3  |-  A. x  e.  ~P  A A. y  e.  ~P  A ( x  i^i  y )  e. 
~P A
2221a1i 9 . 2  |-  ( A  e.  V  ->  A. x  e.  ~P  A A. y  e.  ~P  A ( x  i^i  y )  e. 
~P A )
23 pwexg 4099 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
24 istopg 12155 . . 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 ) ) )
2523, 24syl 14 . 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 ) ) )
268, 22, 25mpbir2and 928 1  |-  ( A  e.  V  ->  ~P A  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329    e. wcel 1480   A.wral 2414   _Vcvv 2681    i^i cin 3065    C_ wss 3066   ~Pcpw 3505   U.cuni 3731   Topctop 12153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-uni 3732  df-top 12154
This theorem is referenced by:  topnex  12244  distopon  12245  distps  12249  discld  12294  restdis  12342  txdis  12435
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