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Theorem dishaus 17327
Description: A discrete topology is Hausdorff. Morris, Topology without tears, p.72, ex. 13. (Contributed by FL, 24-Jun-2007.) (Proof shortened by Mario Carneiro, 8-Apr-2015.)
Assertion
Ref Expression
dishaus  |-  ( A  e.  V  ->  ~P A  e.  Haus )

Proof of Theorem dishaus
Dummy variables  v  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 distop 16950 . 2  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 simplrl 736 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  x  e.  A )
32snssd 3858 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { x }  C_  A )
4 snex 4318 . . . . . . 7  |-  { x }  e.  _V
54elpw 3720 . . . . . 6  |-  ( { x }  e.  ~P A 
<->  { x }  C_  A )
63, 5sylibr 203 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { x }  e.  ~P A
)
7 simplrr 737 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  y  e.  A )
87snssd 3858 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { y }  C_  A )
9 snex 4318 . . . . . . 7  |-  { y }  e.  _V
109elpw 3720 . . . . . 6  |-  ( { y }  e.  ~P A 
<->  { y }  C_  A )
118, 10sylibr 203 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { y }  e.  ~P A
)
12 vex 2876 . . . . . . 7  |-  x  e. 
_V
1312snid 3756 . . . . . 6  |-  x  e. 
{ x }
1413a1i 10 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  x  e.  { x } )
15 vex 2876 . . . . . . 7  |-  y  e. 
_V
1615snid 3756 . . . . . 6  |-  y  e. 
{ y }
1716a1i 10 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  y  e.  { y } )
18 disjsn2 3784 . . . . . 6  |-  ( x  =/=  y  ->  ( { x }  i^i  { y } )  =  (/) )
1918adantl 452 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  ( { x }  i^i  { y } )  =  (/) )
20 eleq2 2427 . . . . . . 7  |-  ( u  =  { x }  ->  ( x  e.  u  <->  x  e.  { x }
) )
21 ineq1 3451 . . . . . . . 8  |-  ( u  =  { x }  ->  ( u  i^i  v
)  =  ( { x }  i^i  v
) )
2221eqeq1d 2374 . . . . . . 7  |-  ( u  =  { x }  ->  ( ( u  i^i  v )  =  (/)  <->  ( { x }  i^i  v )  =  (/) ) )
2320, 223anbi13d 1255 . . . . . 6  |-  ( u  =  { x }  ->  ( ( x  e.  u  /\  y  e.  v  /\  ( u  i^i  v )  =  (/) )  <->  ( x  e. 
{ x }  /\  y  e.  v  /\  ( { x }  i^i  v )  =  (/) ) ) )
24 eleq2 2427 . . . . . . 7  |-  ( v  =  { y }  ->  ( y  e.  v  <->  y  e.  {
y } ) )
25 ineq2 3452 . . . . . . . 8  |-  ( v  =  { y }  ->  ( { x }  i^i  v )  =  ( { x }  i^i  { y } ) )
2625eqeq1d 2374 . . . . . . 7  |-  ( v  =  { y }  ->  ( ( { x }  i^i  v
)  =  (/)  <->  ( {
x }  i^i  {
y } )  =  (/) ) )
2724, 263anbi23d 1256 . . . . . 6  |-  ( v  =  { y }  ->  ( ( x  e.  { x }  /\  y  e.  v  /\  ( { x }  i^i  v )  =  (/) ) 
<->  ( x  e.  {
x }  /\  y  e.  { y }  /\  ( { x }  i^i  { y } )  =  (/) ) ) )
2823, 27rspc2ev 2977 . . . . 5  |-  ( ( { x }  e.  ~P A  /\  { y }  e.  ~P A  /\  ( x  e.  {
x }  /\  y  e.  { y }  /\  ( { x }  i^i  { y } )  =  (/) ) )  ->  E. u  e.  ~P  A E. v  e.  ~P  A ( x  e.  u  /\  y  e.  v  /\  (
u  i^i  v )  =  (/) ) )
296, 11, 14, 17, 19, 28syl113anc 1195 . . . 4  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  E. u  e.  ~P  A E. v  e.  ~P  A ( x  e.  u  /\  y  e.  v  /\  (
u  i^i  v )  =  (/) ) )
3029ex 423 . . 3  |-  ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  =/=  y  ->  E. u  e.  ~P  A E. v  e.  ~P  A ( x  e.  u  /\  y  e.  v  /\  ( u  i^i  v )  =  (/) ) ) )
3130ralrimivva 2720 . 2  |-  ( A  e.  V  ->  A. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  E. u  e.  ~P  A E. v  e.  ~P  A ( x  e.  u  /\  y  e.  v  /\  (
u  i^i  v )  =  (/) ) ) )
32 unipw 4327 . . . 4  |-  U. ~P A  =  A
3332eqcomi 2370 . . 3  |-  A  = 
U. ~P A
3433ishaus 17267 . 2  |-  ( ~P A  e.  Haus  <->  ( ~P A  e.  Top  /\  A. x  e.  A  A. y  e.  A  (
x  =/=  y  ->  E. u  e.  ~P  A E. v  e.  ~P  A ( x  e.  u  /\  y  e.  v  /\  ( u  i^i  v )  =  (/) ) ) ) )
351, 31, 34sylanbrc 645 1  |-  ( A  e.  V  ->  ~P A  e.  Haus )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   E.wrex 2629    i^i cin 3237    C_ wss 3238   (/)c0 3543   ~Pcpw 3714   {csn 3729   U.cuni 3929   Topctop 16848   Hauscha 17253
This theorem is referenced by:  ssoninhaus  25714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-pw 3716  df-sn 3735  df-pr 3736  df-uni 3930  df-top 16853  df-haus 17260
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