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Theorem ist0 18943
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 18968. (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 4118 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
2 ist0.1 . . . 4  |-  X  = 
U. J
31, 2syl6eqr 2493 . . 3  |-  ( j  =  J  ->  U. j  =  X )
4 raleq 2936 . . . . 5  |-  ( j  =  J  ->  ( A. o  e.  j 
( x  e.  o  <-> 
y  e.  o )  <->  A. o  e.  J  ( x  e.  o  <->  y  e.  o ) ) )
54imbi1d 317 . . . 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 2950 . . 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 2950 . 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 18936 . 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 3138 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2734   U.cuni 4110   Topctop 18517   Kol2ct0 18929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-uni 4111  df-t0 18936
This theorem is referenced by:  t0sep  18947  t0top  18952  ist0-2  18967  cnt0  18969
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