MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  t0dist Unicode version

Theorem t0dist 17053
Description: Any two distinct points in a T0 space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
ist0.1  |-  X  = 
U. J
Assertion
Ref Expression
t0dist  |-  ( ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B ) )  ->  E. o  e.  J  -.  ( A  e.  o  <-> 
B  e.  o ) )
Distinct variable groups:    A, o    B, o    o, J    o, X

Proof of Theorem t0dist
StepHypRef Expression
1 ist0.1 . . . . . 6  |-  X  = 
U. J
21t0sep 17052 . . . . 5  |-  ( ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A. o  e.  J  ( A  e.  o  <->  B  e.  o )  ->  A  =  B ) )
32necon3ad 2482 . . . 4  |-  ( ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A  =/=  B  ->  -.  A. o  e.  J  ( A  e.  o  <->  B  e.  o
) ) )
43exp32 588 . . 3  |-  ( J  e.  Kol2  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( A  =/=  B  ->  -.  A. o  e.  J  ( A  e.  o  <->  B  e.  o ) ) ) ) )
543imp2 1166 . 2  |-  ( ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B ) )  ->  -.  A. o  e.  J  ( A  e.  o  <->  B  e.  o
) )
6 rexnal 2554 . 2  |-  ( E. o  e.  J  -.  ( A  e.  o  <->  B  e.  o )  <->  -.  A. o  e.  J  ( A  e.  o  <->  B  e.  o
) )
75, 6sylibr 203 1  |-  ( ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B ) )  ->  E. o  e.  J  -.  ( A  e.  o  <-> 
B  e.  o ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   U.cuni 3827   Kol2ct0 17034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-uni 3828  df-t0 17041
  Copyright terms: Public domain W3C validator