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Theorem ist0 17048
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 17073. (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 3836 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
2 ist0.1 . . . 4  |-  X  = 
U. J
31, 2syl6eqr 2333 . . 3  |-  ( j  =  J  ->  U. j  =  X )
4 raleq 2736 . . . . 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 2748 . . 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 2748 . 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 17041 . 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 2925 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 1623    e. wcel 1684   A.wral 2543   U.cuni 3827   Topctop 16631   Kol2ct0 17034
This theorem is referenced by:  t0sep  17052  t0top  17057  ist0-2  17072  cnt0  17074
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-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-uni 3828  df-t0 17041
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