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Theorem eltrans 24431
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 24398 . . 3  |-  Trans  =  ( _V  \  ran  (
(  _E  o.  _E  )  \  _E  ) )
21eleq2i 2347 . 2  |-  ( A  e.  Trans 
<->  A  e.  ( _V 
\  ran  ( (  _E  o.  _E  )  \  _E  ) ) )
3 eltrans.1 . . 3  |-  A  e. 
_V
43dftr6 24107 . 2  |-  ( Tr  A  <->  A  e.  ( _V  \  ran  ( (  _E  o.  _E  )  \  _E  ) )
)
52, 4bitr4i 243 1  |-  ( A  e.  Trans 
<->  Tr  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684   _Vcvv 2788    \ cdif 3149   Tr wtr 4113    _E cep 4303   ran crn 4690    o. ccom 4693   Transctrans 24376
This theorem is referenced by:  dfon3  24432
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-trans 24398
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