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Theorem dftr3 4240
Description: An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)
Assertion
Ref Expression
dftr3  |-  ( Tr  A  <->  A. x  e.  A  x  C_  A )
Distinct variable group:    x, A

Proof of Theorem dftr3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dftr5 4239 . 2  |-  ( Tr  A  <->  A. x  e.  A  A. y  e.  x  y  e.  A )
2 dfss3 3274 . . 3  |-  ( x 
C_  A  <->  A. y  e.  x  y  e.  A )
32ralbii 2666 . 2  |-  ( A. x  e.  A  x  C_  A  <->  A. x  e.  A  A. y  e.  x  y  e.  A )
41, 3bitr4i 244 1  |-  ( Tr  A  <->  A. x  e.  A  x  C_  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1717   A.wral 2642    C_ wss 3256   Tr wtr 4236
This theorem is referenced by:  trss  4245  trin  4246  triun  4249  trint  4251  tron  4538  ssorduni  4699  suceloni  4726  ordtypelem2  7414  tcwf  7733  itunitc  8227  wunex2  8539  wfgru  8617  tfrALTlem  25293
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-v 2894  df-in 3263  df-ss 3270  df-uni 3951  df-tr 4237
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