MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dftr4 Unicode version

Theorem dftr4 4248
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 4244 . 2  |-  ( Tr  A  <->  U. A  C_  A
)
2 sspwuni 4117 . 2  |-  ( A 
C_  ~P A  <->  U. A  C_  A )
31, 2bitr4i 244 1  |-  ( Tr  A  <->  A  C_  ~P A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    C_ wss 3263   ~Pcpw 3742   U.cuni 3957   Tr wtr 4243
This theorem is referenced by:  tr0  4254  pwtr  4357  r1ordg  7637  r1sssuc  7642  r1val1  7645  ackbij2lem3  8054  tsktrss  8569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-v 2901  df-in 3270  df-ss 3277  df-pw 3744  df-uni 3958  df-tr 4244
  Copyright terms: Public domain W3C validator