MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-tr Structured version   Unicode version

Definition df-tr 4334
Description: Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5281). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4335 (which is suggestive of the word "transitive"), dftr3 4337, dftr4 4338, dftr5 4336, and (when  A is a set) unisuc 4692. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)
Assertion
Ref Expression
df-tr  |-  ( Tr  A  <->  U. A  C_  A
)

Detailed syntax breakdown of Definition df-tr
StepHypRef Expression
1 cA . . 3  class  A
21wtr 4333 . 2  wff  Tr  A
31cuni 4044 . . 3  class  U. A
43, 1wss 3309 . 2  wff  U. A  C_  A
52, 4wb 178 1  wff  ( Tr  A  <->  U. A  C_  A
)
Colors of variables: wff set class
This definition is referenced by:  dftr2  4335  dftr4  4338  treq  4339  trv  4345  pwtr  4451  unisuc  4692  orduniss  4711  onuninsuci  4855  trcl  7700  tc2  7717  r1tr2  7739  tskuni  8696  untangtr  25198  hfuni  26160
  Copyright terms: Public domain W3C validator