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Theorem dftr3 4057
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 4056 . 2  |-  ( Tr  A  <->  A. x  e.  A  A. y  e.  x  y  e.  A )
2 dfss3 3112 . . 3  |-  ( x 
C_  A  <->  A. y  e.  x  y  e.  A )
32ralbii 2538 . 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 2516    C_ wss 3094   Tr wtr 4053
This theorem is referenced by:  trss  4062  trin  4063  triun  4066  trint  4068  tron  4352  ssorduni  4514  suceloni  4541  ordtypelem2  7167  tcwf  7486  itunitc  7980  wunex2  8293  wfgru  8371  gruina  8373  grur1  8375  tfrALTlem  23610  celsor  24442
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 1927  ax-ext 2237
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 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2520  df-v 2742  df-in 3101  df-ss 3108  df-uni 3769  df-tr 4054
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