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Theorem eltrans 25729
Description: Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
Hypothesis
Ref Expression
eltrans.1  |-  A  e. 
_V
Assertion
Ref Expression
eltrans  |-  ( A  e.  Trans 
<->  Tr  A )

Proof of Theorem eltrans
StepHypRef Expression
1 df-trans 25694 . . 3  |-  Trans  =  ( _V  \  ran  (
(  _E  o.  _E  )  \  _E  ) )
21eleq2i 2500 . 2  |-  ( A  e.  Trans 
<->  A  e.  ( _V 
\  ran  ( (  _E  o.  _E  )  \  _E  ) ) )
3 eltrans.1 . . 3  |-  A  e. 
_V
43dftr6 25366 . 2  |-  ( Tr  A  <->  A  e.  ( _V  \  ran  ( (  _E  o.  _E  )  \  _E  ) )
)
52, 4bitr4i 244 1  |-  ( A  e.  Trans 
<->  Tr  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1725   _Vcvv 2949    \ cdif 3310   Tr wtr 4295    _E cep 4485   ran crn 4872    o. ccom 4875   Transctrans 25670
This theorem is referenced by:  dfon3  25730
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-tr 4296  df-eprel 4487  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-trans 25694
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