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Theorem dftr3 4014
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
StepHypRef Expression
1 dftr5 4013 . 2  |-  ( Tr  A  <->  A. x  e.  A  A. y  e.  x  y  e.  A )
2 dfss3 3093 . . 3  |-  ( x 
C_  A  <->  A. y  e.  x  y  e.  A )
32ralbii 2531 . 2  |-  ( A. x  e.  A  x  C_  A  <->  A. x  e.  A  A. y  e.  x  y  e.  A )
41, 3bitr4i 245 1  |-  ( Tr  A  <->  A. x  e.  A  x  C_  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1621   A.wral 2509    C_ wss 3078   Tr wtr 4010
This theorem is referenced by:  trss  4019  trin  4020  triun  4023  trint  4025  tron  4308  ssorduni  4468  suceloni  4495  ordtypelem2  7118  tcwf  7437  itunitc  7931  wunex2  8240  wfgru  8318  gruina  8320  grur1  8322  tfrALTlem  23444  celsor  24276
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-v 2729  df-in 3085  df-ss 3089  df-uni 3728  df-tr 4011
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