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Theorem ist0 17376
 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 17401. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
ist0.1
Assertion
Ref Expression
ist0
Distinct variable groups:   ,,,   ,,,

Proof of Theorem ist0
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 unieq 4016 . . . 4
2 ist0.1 . . . 4
31, 2syl6eqr 2485 . . 3
4 raleq 2896 . . . . 5
54imbi1d 309 . . . 4
63, 5raleqbidv 2908 . . 3
73, 6raleqbidv 2908 . 2
8 df-t0 17369 . 2
97, 8elrab2 3086 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  cuni 4007  ctop 16950  ct0 17362 This theorem is referenced by:  t0sep  17380  t0top  17385  ist0-2  17400  cnt0  17402 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-uni 4008  df-t0 17369
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