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Theorem dftr3 4119
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 4118 . 2  |-  ( Tr  A  <->  A. x  e.  A  A. y  e.  x  y  e.  A )
2 dfss3 3172 . . 3  |-  ( x 
C_  A  <->  A. y  e.  x  y  e.  A )
32ralbii 2569 . 2  |-  ( A. x  e.  A  x  C_  A  <->  A. x  e.  A  A. y  e.  x  y  e.  A )
41, 3bitr4i 243 1  |-  ( Tr  A  <->  A. x  e.  A  x  C_  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1686   A.wral 2545    C_ wss 3154   Tr wtr 4115
This theorem is referenced by:  trss  4124  trin  4125  triun  4128  trint  4130  tron  4417  ssorduni  4579  suceloni  4606  ordtypelem2  7236  tcwf  7555  itunitc  8049  wunex2  8362  wfgru  8440  gruina  8442  grur1  8444  tfrALTlem  24278  celsor  25122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-v 2792  df-in 3161  df-ss 3168  df-uni 3830  df-tr 4116
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