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Theorem dftr4 4388
Description: An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
Assertion
Ref Expression
dftr4  |-  ( Tr  A  <->  A  C_  ~P A
)

Proof of Theorem dftr4
StepHypRef Expression
1 df-tr 4384 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 4254 . 2  |-  ( A 
C_  ~P A  <->  U. A  C_  A )
31, 2bitr4i 252 1  |-  ( Tr  A  <->  A  C_  ~P A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    C_ wss 3326   ~Pcpw 3858   U.cuni 4089   Tr wtr 4383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-v 2972  df-in 3333  df-ss 3340  df-pw 3860  df-uni 4090  df-tr 4384
This theorem is referenced by:  tr0  4394  pwtr  4543  r1ordg  7983  r1sssuc  7988  r1val1  7991  ackbij2lem3  8408  tsktrss  8926
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