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Theorem dishaus 17408
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 17023 . 2  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 simplrl 737 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  x  e.  A )
32snssd 3911 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { x }  C_  A )
4 snex 4373 . . . . . . 7  |-  { x }  e.  _V
54elpw 3773 . . . . . 6  |-  ( { x }  e.  ~P A 
<->  { x }  C_  A )
63, 5sylibr 204 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { x }  e.  ~P A
)
7 simplrr 738 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  y  e.  A )
87snssd 3911 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { y }  C_  A )
9 snex 4373 . . . . . . 7  |-  { y }  e.  _V
109elpw 3773 . . . . . 6  |-  ( { y }  e.  ~P A 
<->  { y }  C_  A )
118, 10sylibr 204 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  { y }  e.  ~P A
)
12 vex 2927 . . . . . . 7  |-  x  e. 
_V
1312snid 3809 . . . . . 6  |-  x  e. 
{ x }
1413a1i 11 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  x  e.  { x } )
15 vex 2927 . . . . . . 7  |-  y  e. 
_V
1615snid 3809 . . . . . 6  |-  y  e. 
{ y }
1716a1i 11 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  y  e.  { y } )
18 disjsn2 3837 . . . . . 6  |-  ( x  =/=  y  ->  ( { x }  i^i  { y } )  =  (/) )
1918adantl 453 . . . . 5  |-  ( ( ( A  e.  V  /\  ( x  e.  A  /\  y  e.  A
) )  /\  x  =/=  y )  ->  ( { x }  i^i  { y } )  =  (/) )
20 eleq2 2473 . . . . . . 7  |-  ( u  =  { x }  ->  ( x  e.  u  <->  x  e.  { x }
) )
21 ineq1 3503 . . . . . . . 8  |-  ( u  =  { x }  ->  ( u  i^i  v
)  =  ( { x }  i^i  v
) )
2221eqeq1d 2420 . . . . . . 7  |-  ( u  =  { x }  ->  ( ( u  i^i  v )  =  (/)  <->  ( { x }  i^i  v )  =  (/) ) )
2320, 223anbi13d 1256 . . . . . 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 2473 . . . . . . 7  |-  ( v  =  { y }  ->  ( y  e.  v  <->  y  e.  {
y } ) )
25 ineq2 3504 . . . . . . . 8  |-  ( v  =  { y }  ->  ( { x }  i^i  v )  =  ( { x }  i^i  { y } ) )
2625eqeq1d 2420 . . . . . . 7  |-  ( v  =  { y }  ->  ( ( { x }  i^i  v
)  =  (/)  <->  ( {
x }  i^i  {
y } )  =  (/) ) )
2724, 263anbi23d 1257 . . . . . 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 3028 . . . . 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 1196 . . . 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 424 . . 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 2766 . 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 4382 . . . 4  |-  U. ~P A  =  A
3332eqcomi 2416 . . 3  |-  A  = 
U. ~P A
3433ishaus 17348 . 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 646 1  |-  ( A  e.  V  ->  ~P A  e.  Haus )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675    i^i cin 3287    C_ wss 3288   (/)c0 3596   ~Pcpw 3767   {csn 3782   U.cuni 3983   Topctop 16921   Hauscha 17334
This theorem is referenced by:  ssoninhaus  26110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-pw 3769  df-sn 3788  df-pr 3789  df-uni 3984  df-top 16926  df-haus 17341
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