MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isperf3 Unicode version

Theorem isperf3 16984
Description: A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
isperf3  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\ 
A. x  e.  X  -.  { x }  e.  J ) )
Distinct variable groups:    x, J    x, X

Proof of Theorem isperf3
StepHypRef Expression
1 lpfval.1 . . 3  |-  X  = 
U. J
21isperf2 16983 . 2  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\  X  C_  ( ( limPt `  J ) `  X ) ) )
3 dfss3 3246 . . . 4  |-  ( X 
C_  ( ( limPt `  J ) `  X
)  <->  A. x  e.  X  x  e.  ( ( limPt `  J ) `  X ) )
41maxlp 16978 . . . . . 6  |-  ( J  e.  Top  ->  (
x  e.  ( (
limPt `  J ) `  X )  <->  ( x  e.  X  /\  -.  {
x }  e.  J
) ) )
54baibd 875 . . . . 5  |-  ( ( J  e.  Top  /\  x  e.  X )  ->  ( x  e.  ( ( limPt `  J ) `  X )  <->  -.  { x }  e.  J )
)
65ralbidva 2635 . . . 4  |-  ( J  e.  Top  ->  ( A. x  e.  X  x  e.  ( ( limPt `  J ) `  X )  <->  A. x  e.  X  -.  { x }  e.  J )
)
73, 6syl5bb 248 . . 3  |-  ( J  e.  Top  ->  ( X  C_  ( ( limPt `  J ) `  X
)  <->  A. x  e.  X  -.  { x }  e.  J ) )
87pm5.32i 618 . 2  |-  ( ( J  e.  Top  /\  X  C_  ( ( limPt `  J ) `  X
) )  <->  ( J  e.  Top  /\  A. x  e.  X  -.  { x }  e.  J )
)
92, 8bitri 240 1  |-  ( J  e. Perf 
<->  ( J  e.  Top  /\ 
A. x  e.  X  -.  { x }  e.  J ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619    C_ wss 3228   {csn 3716   U.cuni 3906   ` cfv 5334   Topctop 16731   limPtclp 16966  Perfcperf 16967
This theorem is referenced by:  perfi  16986  perfopn  17015  t1conperf  17262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-top 16736  df-cld 16856  df-ntr 16857  df-cls 16858  df-lp 16968  df-perf 16969
  Copyright terms: Public domain W3C validator