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Theorem dftr3 4118
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 4117 . 2  |-  ( Tr  A  <->  A. x  e.  A  A. y  e.  x  y  e.  A )
2 dfss3 3171 . . 3  |-  ( x 
C_  A  <->  A. y  e.  x  y  e.  A )
32ralbii 2568 . 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 1685   A.wral 2544    C_ wss 3153   Tr wtr 4114
This theorem is referenced by:  trss  4123  trin  4124  triun  4127  trint  4129  tron  4414  ssorduni  4576  suceloni  4603  ordtypelem2  7230  tcwf  7549  itunitc  8043  wunex2  8356  wfgru  8434  gruina  8436  grur1  8438  tfrALTlem  23680  celsor  24521
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 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-v 2791  df-in 3160  df-ss 3167  df-uni 3829  df-tr 4115
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