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Theorem ist0 17064
Description: The predicate "is a T0 space." Every pair of distinct points is topologically distinguishable. For the way this definition is usually encountered, see ist0-3 17089. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
ist0.1  |-  X  = 
U. J
Assertion
Ref Expression
ist0  |-  ( J  e.  Kol2  <->  ( J  e. 
Top  /\  A. x  e.  X  A. y  e.  X  ( A. o  e.  J  (
x  e.  o  <->  y  e.  o )  ->  x  =  y ) ) )
Distinct variable groups:    x, o,
y, J    o, X, x, y

Proof of Theorem ist0
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 unieq 3852 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
2 ist0.1 . . . 4  |-  X  = 
U. J
31, 2syl6eqr 2346 . . 3  |-  ( j  =  J  ->  U. j  =  X )
4 raleq 2749 . . . . 5  |-  ( j  =  J  ->  ( A. o  e.  j 
( x  e.  o  <-> 
y  e.  o )  <->  A. o  e.  J  ( x  e.  o  <->  y  e.  o ) ) )
54imbi1d 308 . . . 4  |-  ( j  =  J  ->  (
( A. o  e.  j  ( x  e.  o  <->  y  e.  o )  ->  x  =  y )  <->  ( A. o  e.  J  (
x  e.  o  <->  y  e.  o )  ->  x  =  y ) ) )
63, 5raleqbidv 2761 . . 3  |-  ( j  =  J  ->  ( A. y  e.  U. j
( A. o  e.  j  ( x  e.  o  <->  y  e.  o )  ->  x  =  y )  <->  A. y  e.  X  ( A. o  e.  J  (
x  e.  o  <->  y  e.  o )  ->  x  =  y ) ) )
73, 6raleqbidv 2761 . 2  |-  ( j  =  J  ->  ( A. x  e.  U. j A. y  e.  U. j
( A. o  e.  j  ( x  e.  o  <->  y  e.  o )  ->  x  =  y )  <->  A. x  e.  X  A. y  e.  X  ( A. o  e.  J  (
x  e.  o  <->  y  e.  o )  ->  x  =  y ) ) )
8 df-t0 17057 . 2  |-  Kol2  =  { j  e.  Top  | 
A. x  e.  U. j A. y  e.  U. j ( A. o  e.  j  ( x  e.  o  <->  y  e.  o )  ->  x  =  y ) }
97, 8elrab2 2938 1  |-  ( J  e.  Kol2  <->  ( J  e. 
Top  /\  A. x  e.  X  A. y  e.  X  ( A. o  e.  J  (
x  e.  o  <->  y  e.  o )  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   U.cuni 3843   Topctop 16647   Kol2ct0 17050
This theorem is referenced by:  t0sep  17068  t0top  17073  ist0-2  17088  cnt0  17090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-uni 3844  df-t0 17057
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