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Definition df-ps 14322
Description: Define the class of all posets (partially ordered sets) with weak ordering (e.g. "less than or equal to" instead of "less than"). A poset is a relation which is transitive, reflexive, and antisymmetric. (Contributed by NM, 11-May-2008.)
Assertion
Ref Expression
df-ps  |-  PosetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) ) }

Detailed syntax breakdown of Definition df-ps
StepHypRef Expression
1 cps 14317 . 2  class  PosetRel
2 vr . . . . . 6  set  r
32cv 1631 . . . . 5  class  r
43wrel 4710 . . . 4  wff  Rel  r
53, 3ccom 4709 . . . . 5  class  ( r  o.  r )
65, 3wss 3165 . . . 4  wff  ( r  o.  r )  C_  r
73ccnv 4704 . . . . . 6  class  `' r
83, 7cin 3164 . . . . 5  class  ( r  i^i  `' r )
9 cid 4320 . . . . . 6  class  _I
103cuni 3843 . . . . . . 7  class  U. r
1110cuni 3843 . . . . . 6  class  U. U. r
129, 11cres 4707 . . . . 5  class  (  _I  |`  U. U. r )
138, 12wceq 1632 . . . 4  wff  ( r  i^i  `' r )  =  (  _I  |`  U. U. r )
144, 6, 13w3a 934 . . 3  wff  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) )
1514, 2cab 2282 . 2  class  { r  |  ( Rel  r  /\  ( r  o.  r
)  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r
) ) }
161, 15wceq 1632 1  wff  PosetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) ) }
Colors of variables: wff set class
This definition is referenced by:  isps  14327  posprsr  25343  dfps2  25392
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