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Theorem ist0 19988
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 20013. (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 4243 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
2 ist0.1 . . . 4  |-  X  = 
U. J
31, 2syl6eqr 2513 . . 3  |-  ( j  =  J  ->  U. j  =  X )
4 raleq 3051 . . . . 5  |-  ( j  =  J  ->  ( A. o  e.  j 
( x  e.  o  <-> 
y  e.  o )  <->  A. o  e.  J  ( x  e.  o  <->  y  e.  o ) ) )
54imbi1d 315 . . . 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 3065 . . 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 3065 . 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 19981 . 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 3256 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   U.cuni 4235   Topctop 19561   Kol2ct0 19974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-uni 4236  df-t0 19981
This theorem is referenced by:  t0sep  19992  t0top  19997  ist0-2  20012  cnt0  20014
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