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Definition df-preset 14062
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 14060 . 2  class  Preset
2 vx . . . . . . . . . . 11  set  x
32cv 1622 . . . . . . . . . 10  class  x
4 vr . . . . . . . . . . 11  set  r
54cv 1622 . . . . . . . . . 10  class  r
63, 3, 5wbr 4023 . . . . . . . . 9  wff  x r x
7 vy . . . . . . . . . . . . 13  set  y
87cv 1622 . . . . . . . . . . . 12  class  y
93, 8, 5wbr 4023 . . . . . . . . . . 11  wff  x r y
10 vz . . . . . . . . . . . . 13  set  z
1110cv 1622 . . . . . . . . . . . 12  class  z
128, 11, 5wbr 4023 . . . . . . . . . . 11  wff  y r z
139, 12wa 358 . . . . . . . . . 10  wff  ( x r y  /\  y
r z )
143, 11, 5wbr 4023 . . . . . . . . . 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 1622 . . . . . . . 8  class  b
1916, 10, 18wral 2543 . . . . . . 7  wff  A. z  e.  b  ( x
r x  /\  (
( x r y  /\  y r z )  ->  x r
z ) )
2019, 7, 18wral 2543 . . . . . 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 2543 . . . . 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 1622 . . . . . 6  class  f
24 cple 13215 . . . . . 6  class  le
2523, 24cfv 5255 . . . . 5  class  ( le
`  f )
2621, 4, 25wsbc 2991 . . . 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 13148 . . . . 5  class  Base
2823, 27cfv 5255 . . . 4  class  ( Base `  f )
2926, 17, 28wsbc 2991 . . 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 2269 . 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 1623 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  14064
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