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Theorem eltrans 25448
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 25415 . . 3  |-  Trans  =  ( _V  \  ran  (
(  _E  o.  _E  )  \  _E  ) )
21eleq2i 2444 . 2  |-  ( A  e.  Trans 
<->  A  e.  ( _V 
\  ran  ( (  _E  o.  _E  )  \  _E  ) ) )
3 eltrans.1 . . 3  |-  A  e. 
_V
43dftr6 25124 . 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 1717   _Vcvv 2892    \ cdif 3253   Tr wtr 4236    _E cep 4426   ran crn 4812    o. ccom 4815   Transctrans 25393
This theorem is referenced by:  dfon3  25449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-trans 25415
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