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Theorem dishaus 19008
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 18622 . 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 4039 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { x }  C_  A )
4 snex 4554 . . . . . . 7  |-  { x }  e.  _V
54elpw 3887 . . . . . 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 4039 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { y }  C_  A )
9 snex 4554 . . . . . . 7  |-  { y }  e.  _V
109elpw 3887 . . . . . 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 3927 . . . . . 6  |-  x  e. 
{ x }
1312a1i 11 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  x  e.  { x } )
14 ssnid 3927 . . . . . 6  |-  y  e. 
{ y }
1514a1i 11 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  y  e.  { y } )
16 disjsn2 3958 . . . . . 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 2504 . . . . . . 7  |-  ( u  =  { x }  ->  ( x  e.  u  <->  x  e.  { x }
) )
19 ineq1 3566 . . . . . . . 8  |-  ( u  =  { x }  ->  ( u  i^i  v
)  =  ( { x }  i^i  v
) )
2019eqeq1d 2451 . . . . . . 7  |-  ( u  =  { x }  ->  ( ( u  i^i  v )  =  (/)  <->  ( { x }  i^i  v )  =  (/) ) )
2118, 203anbi13d 1291 . . . . . 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 2504 . . . . . . 7  |-  ( v  =  { y }  ->  ( y  e.  v  <->  y  e.  {
y } ) )
23 ineq2 3567 . . . . . . . 8  |-  ( v  =  { y }  ->  ( { x }  i^i  v )  =  ( { x }  i^i  { y } ) )
2423eqeq1d 2451 . . . . . . 7  |-  ( v  =  { y }  ->  ( ( { x }  i^i  v
)  =  (/)  <->  ( {
x }  i^i  {
y } )  =  (/) ) )
2522, 243anbi23d 1292 . . . . . 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 3102 . . . . 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 1230 . . . 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 2829 . 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 4563 . . . 4  |-  U. ~P A  =  A
3130eqcomi 2447 . . 3  |-  A  = 
U. ~P A
3231ishaus 18948 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737    i^i cin 3348    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   {csn 3898   U.cuni 4112   Topctop 18520   Hauscha 18934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-pw 3883  df-sn 3899  df-pr 3901  df-uni 4113  df-top 18525  df-haus 18941
This theorem is referenced by:  ssoninhaus  28316
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