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

Definition df-tr 4267
Description: Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5209). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4268 (which is suggestive of the word "transitive"), dftr3 4270, dftr4 4271, dftr5 4269, and (when  A is a set) unisuc 4621. 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 4266 . 2  wff  Tr  A
31cuni 3979 . . 3  class  U. A
43, 1wss 3284 . 2  wff  U. A  C_  A
52, 4wb 177 1  wff  ( Tr  A  <->  U. A  C_  A
)
Colors of variables: wff set class
This definition is referenced by:  dftr2  4268  dftr4  4271  treq  4272  trv  4278  pwtr  4380  unisuc  4621  orduniss  4639  onuninsuci  4783  trcl  7624  tc2  7641  r1tr2  7663  tskuni  8618  untangtr  25120  hfuni  26033
  Copyright terms: Public domain W3C validator