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Definition df-preset 14078
Description: Define the class of preordered sets (presets). A preset is a set equipped with a transitive and reflexive relation.

Preorders are a natural generalization of order for sets where there is a well-defined ordering, but it in some sense "fails to capture the whole story", in that there may be pairs of elements which are indistinguishable under the order. Two elements which are not equal but are less-or-equal to each other behave the same under all order operations and may be thought of as "tied".

A preorder can naturally be strengthened by requiring that there are no ties, resulting in a partial order, or by stating that all comparable pairs of elements are tied, resulting in an equivalence relation. Every preorder naturally factors into these two types; the tied relation on a preorder is an equivalence relation and the quotient under that relation is a partial order. (Contributed by FL, 17-Nov-2014.) (Revised by Stefan O'Rear, 31-Jan-2015.)

Assertion
Ref Expression
df-preset  |-  Preset  =  {
f  |  [. ( Base `  f )  / 
b ]. [. ( le
`  f )  / 
r ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r z )  ->  x r z ) ) }
Distinct variable group:    f, b, r, x, y, z

Detailed syntax breakdown of Definition df-preset
StepHypRef Expression
1 cpreset 14076 . 2  class  Preset
2 vx . . . . . . . . . . 11  set  x
32cv 1631 . . . . . . . . . 10  class  x
4 vr . . . . . . . . . . 11  set  r
54cv 1631 . . . . . . . . . 10  class  r
63, 3, 5wbr 4039 . . . . . . . . 9  wff  x r x
7 vy . . . . . . . . . . . . 13  set  y
87cv 1631 . . . . . . . . . . . 12  class  y
93, 8, 5wbr 4039 . . . . . . . . . . 11  wff  x r y
10 vz . . . . . . . . . . . . 13  set  z
1110cv 1631 . . . . . . . . . . . 12  class  z
128, 11, 5wbr 4039 . . . . . . . . . . 11  wff  y r z
139, 12wa 358 . . . . . . . . . 10  wff  ( x r y  /\  y
r z )
143, 11, 5wbr 4039 . . . . . . . . . 10  wff  x r z
1513, 14wi 4 . . . . . . . . 9  wff  ( ( x r y  /\  y r z )  ->  x r z )
166, 15wa 358 . . . . . . . 8  wff  ( x r x  /\  (
( x r y  /\  y r z )  ->  x r
z ) )
17 vb . . . . . . . . 9  set  b
1817cv 1631 . . . . . . . 8  class  b
1916, 10, 18wral 2556 . . . . . . 7  wff  A. z  e.  b  ( x
r x  /\  (
( x r y  /\  y r z )  ->  x r
z ) )
2019, 7, 18wral 2556 . . . . . 6  wff  A. y  e.  b  A. z  e.  b  ( x
r x  /\  (
( x r y  /\  y r z )  ->  x r
z ) )
2120, 2, 18wral 2556 . . . . 5  wff  A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x
r x  /\  (
( x r y  /\  y r z )  ->  x r
z ) )
22 vf . . . . . . 7  set  f
2322cv 1631 . . . . . 6  class  f
24 cple 13231 . . . . . 6  class  le
2523, 24cfv 5271 . . . . 5  class  ( le
`  f )
2621, 4, 25wsbc 3004 . . . 4  wff  [. ( le `  f )  / 
r ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r z )  ->  x r z ) )
27 cbs 13164 . . . . 5  class  Base
2823, 27cfv 5271 . . . 4  class  ( Base `  f )
2926, 17, 28wsbc 3004 . . 3  wff  [. ( Base `  f )  / 
b ]. [. ( le
`  f )  / 
r ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r z )  ->  x r z ) )
3029, 22cab 2282 . 2  class  { f  |  [. ( Base `  f )  /  b ]. [. ( le `  f )  /  r ]. A. x  e.  b 
A. y  e.  b 
A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y
r z )  ->  x r z ) ) }
311, 30wceq 1632 1  wff  Preset  =  {
f  |  [. ( Base `  f )  / 
b ]. [. ( le
`  f )  / 
r ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r z )  ->  x r z ) ) }
Colors of variables: wff set class
This definition is referenced by:  isprs  14080
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