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Theorem dftr4 4058
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 4054 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 3928 . 2  |-  ( A 
C_  ~P A  <->  U. A  C_  A )
31, 2bitr4i 245 1  |-  ( Tr  A  <->  A  C_  ~P A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    C_ wss 3094   ~Pcpw 3566   U.cuni 3768   Tr wtr 4053
This theorem is referenced by:  tr0  4064  pwtr  4164  r1ordg  7383  r1sssuc  7388  r1val1  7391  ackbij2lem3  7800  tsktrss  8316
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-pw 3568  df-uni 3769  df-tr 4054
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