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Theorem dishaus 19751
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 19365 . 2  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 simplrl 759 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  x  e.  A )
32snssd 4178 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { x }  C_  A )
4 snex 4694 . . . . . . 7  |-  { x }  e.  _V
54elpw 4022 . . . . . 6  |-  ( { x }  e.  ~P A 
<->  { x }  C_  A )
63, 5sylibr 212 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { x }  e.  ~P A
)
7 simplrr 760 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  y  e.  A )
87snssd 4178 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { y }  C_  A )
9 snex 4694 . . . . . . 7  |-  { y }  e.  _V
109elpw 4022 . . . . . 6  |-  ( { y }  e.  ~P A 
<->  { y }  C_  A )
118, 10sylibr 212 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { y }  e.  ~P A
)
12 ssnid 4062 . . . . . 6  |-  x  e. 
{ x }
1312a1i 11 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  x  e.  { x } )
14 ssnid 4062 . . . . . 6  |-  y  e. 
{ y }
1514a1i 11 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  y  e.  { y } )
16 disjsn2 4095 . . . . . 6  |-  ( x  =/=  y  ->  ( { x }  i^i  { y } )  =  (/) )
1716adantl 466 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  ( { x }  i^i  { y } )  =  (/) )
18 eleq2 2540 . . . . . . 7  |-  ( u  =  { x }  ->  ( x  e.  u  <->  x  e.  { x }
) )
19 ineq1 3698 . . . . . . . 8  |-  ( u  =  { x }  ->  ( u  i^i  v
)  =  ( { x }  i^i  v
) )
2019eqeq1d 2469 . . . . . . 7  |-  ( u  =  { x }  ->  ( ( u  i^i  v )  =  (/)  <->  ( { x }  i^i  v )  =  (/) ) )
2118, 203anbi13d 1301 . . . . . 6  |-  ( u  =  { x }  ->  ( ( x  e.  u  /\  y  e.  v  /\  ( u  i^i  v )  =  (/) )  <->  ( x  e. 
{ x }  /\  y  e.  v  /\  ( { x }  i^i  v )  =  (/) ) ) )
22 eleq2 2540 . . . . . . 7  |-  ( v  =  { y }  ->  ( y  e.  v  <->  y  e.  {
y } ) )
23 ineq2 3699 . . . . . . . 8  |-  ( v  =  { y }  ->  ( { x }  i^i  v )  =  ( { x }  i^i  { y } ) )
2423eqeq1d 2469 . . . . . . 7  |-  ( v  =  { y }  ->  ( ( { x }  i^i  v
)  =  (/)  <->  ( {
x }  i^i  {
y } )  =  (/) ) )
2522, 243anbi23d 1302 . . . . . 6  |-  ( v  =  { y }  ->  ( ( x  e.  { x }  /\  y  e.  v  /\  ( { x }  i^i  v )  =  (/) ) 
<->  ( x  e.  {
x }  /\  y  e.  { y }  /\  ( { x }  i^i  { y } )  =  (/) ) ) )
2621, 25rspc2ev 3230 . . . . 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 )  =  (/) ) )
276, 11, 13, 15, 17, 26syl113anc 1240 . . . 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 )  =  (/) ) )
2827ex 434 . . 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 )  =  (/) ) ) )
2928ralrimivva 2888 . 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 )  =  (/) ) ) )
30 unipw 4703 . . . 4  |-  U. ~P A  =  A
3130eqcomi 2480 . . 3  |-  A  = 
U. ~P A
3231ishaus 19691 . 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 )  =  (/) ) ) ) )
331, 29, 32sylanbrc 664 1  |-  ( A  e.  V  ->  ~P A  e.  Haus )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   {csn 4033   U.cuni 4251   Topctop 19263   Hauscha 19677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-pw 4018  df-sn 4034  df-pr 4036  df-uni 4252  df-top 19268  df-haus 19684
This theorem is referenced by:  ssoninhaus  29840
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