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Theorem dftr3 4247
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 4246 . 2  |-  ( Tr  A  <->  A. x  e.  A  A. y  e.  x  y  e.  A )
2 dfss3 3281 . . 3  |-  ( x 
C_  A  <->  A. y  e.  x  y  e.  A )
32ralbii 2673 . 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 2649    C_ wss 3263   Tr wtr 4243
This theorem is referenced by:  trss  4252  trin  4253  triun  4256  trint  4258  tron  4545  ssorduni  4706  suceloni  4733  ordtypelem2  7421  tcwf  7740  itunitc  8234  wunex2  8546  wfgru  8624  tfrALTlem  25300
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 2368
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 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-v 2901  df-in 3270  df-ss 3277  df-uni 3958  df-tr 4244
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