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Definition df-preset 14373
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 14371 . 2  class  Preset
2 vx . . . . . . . . . . 11  set  x
32cv 1651 . . . . . . . . . 10  class  x
4 vr . . . . . . . . . . 11  set  r
54cv 1651 . . . . . . . . . 10  class  r
63, 3, 5wbr 4204 . . . . . . . . 9  wff  x r x
7 vy . . . . . . . . . . . . 13  set  y
87cv 1651 . . . . . . . . . . . 12  class  y
93, 8, 5wbr 4204 . . . . . . . . . . 11  wff  x r y
10 vz . . . . . . . . . . . . 13  set  z
1110cv 1651 . . . . . . . . . . . 12  class  z
128, 11, 5wbr 4204 . . . . . . . . . . 11  wff  y r z
139, 12wa 359 . . . . . . . . . 10  wff  ( x r y  /\  y
r z )
143, 11, 5wbr 4204 . . . . . . . . . 10  wff  x r z
1513, 14wi 4 . . . . . . . . 9  wff  ( ( x r y  /\  y r z )  ->  x r z )
166, 15wa 359 . . . . . . . 8  wff  ( x r x  /\  (
( x r y  /\  y r z )  ->  x r
z ) )
17 vb . . . . . . . . 9  set  b
1817cv 1651 . . . . . . . 8  class  b
1916, 10, 18wral 2697 . . . . . . 7  wff  A. z  e.  b  ( x
r x  /\  (
( x r y  /\  y r z )  ->  x r
z ) )
2019, 7, 18wral 2697 . . . . . 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 2697 . . . . 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 1651 . . . . . 6  class  f
24 cple 13524 . . . . . 6  class  le
2523, 24cfv 5445 . . . . 5  class  ( le
`  f )
2621, 4, 25wsbc 3153 . . . 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 13457 . . . . 5  class  Base
2823, 27cfv 5445 . . . 4  class  ( Base `  f )
2926, 17, 28wsbc 3153 . . 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 2421 . 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 1652 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  14375
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