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Theorem dftr4 4495
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 4491 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 4360 . 2  |-  ( A 
C_  ~P A  <->  U. A  C_  A )
31, 2bitr4i 260 1  |-  ( Tr  A  <->  A  C_  ~P A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    C_ wss 3390   ~Pcpw 3942   U.cuni 4190   Tr wtr 4490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-v 3033  df-in 3397  df-ss 3404  df-pw 3944  df-uni 4191  df-tr 4491
This theorem is referenced by:  tr0  4501  pwtr  4653  r1ordg  8267  r1sssuc  8272  r1val1  8275  ackbij2lem3  8689  tsktrss  9204
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