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Theorem t0dist 17391
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 17390 . . . . 5  |-  ( ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A. o  e.  J  ( A  e.  o  <->  B  e.  o )  ->  A  =  B ) )
32necon3ad 2639 . . . 4  |-  ( ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A  =/=  B  ->  -.  A. o  e.  J  ( A  e.  o  <->  B  e.  o
) ) )
43exp32 590 . . 3  |-  ( J  e.  Kol2  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( A  =/=  B  ->  -.  A. o  e.  J  ( A  e.  o  <->  B  e.  o ) ) ) ) )
543imp2 1169 . 2  |-  ( ( J  e.  Kol2  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B ) )  ->  -.  A. o  e.  J  ( A  e.  o  <->  B  e.  o
) )
6 rexnal 2718 . 2  |-  ( E. o  e.  J  -.  ( A  e.  o  <->  B  e.  o )  <->  -.  A. o  e.  J  ( A  e.  o  <->  B  e.  o
) )
75, 6sylibr 205 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   U.cuni 4017   Kol2ct0 17372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-uni 4018  df-t0 17379
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