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Theorem dftr2 4306
Description: An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)
Assertion
Ref Expression
dftr2  |-  ( Tr  A  <->  A. x A. y
( ( x  e.  y  /\  y  e.  A )  ->  x  e.  A ) )
Distinct variable group:    x, y, A

Proof of Theorem dftr2
StepHypRef Expression
1 dfss2 3339 . 2  |-  ( U. A  C_  A  <->  A. x
( x  e.  U. A  ->  x  e.  A
) )
2 df-tr 4305 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
3 19.23v 1915 . . . 4  |-  ( A. y ( ( x  e.  y  /\  y  e.  A )  ->  x  e.  A )  <->  ( E. y ( x  e.  y  /\  y  e.  A )  ->  x  e.  A ) )
4 eluni 4020 . . . . 5  |-  ( x  e.  U. A  <->  E. y
( x  e.  y  /\  y  e.  A
) )
54imbi1i 317 . . . 4  |-  ( ( x  e.  U. A  ->  x  e.  A )  <-> 
( E. y ( x  e.  y  /\  y  e.  A )  ->  x  e.  A ) )
63, 5bitr4i 245 . . 3  |-  ( A. y ( ( x  e.  y  /\  y  e.  A )  ->  x  e.  A )  <->  ( x  e.  U. A  ->  x  e.  A ) )
76albii 1576 . 2  |-  ( A. x A. y ( ( x  e.  y  /\  y  e.  A )  ->  x  e.  A )  <->  A. x ( x  e. 
U. A  ->  x  e.  A ) )
81, 2, 73bitr4i 270 1  |-  ( Tr  A  <->  A. x A. y
( ( x  e.  y  /\  y  e.  A )  ->  x  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551    e. wcel 1726    C_ wss 3322   U.cuni 4017   Tr wtr 4304
This theorem is referenced by:  dftr5  4307  trel  4311  ordelord  4605  suctrALT  4666  suctr  4667  ordom  4856  hartogs  7515  card2on  7524  trcl  7666  tskwe  7839  ondomon  8440  dftr6  25375  elpotr  25410  hftr  26125  dford4  27102  tratrb  28682  trsbc  28687  truniALT  28688  sspwtr  28996  sspwtrALT  28997  sspwtrALT2  28998  pwtrVD  28999  pwtrrVD  29000  suctrALT2VD  29010  suctrALT2  29011  tratrbVD  29035  trsbcVD  29051  truniALTVD  29052  trintALTVD  29054  trintALT  29055  suctrALTcf  29096  suctrALTcfVD  29097  suctrALT3  29098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-ss 3336  df-uni 4018  df-tr 4305
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