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Mirrors > Home > MPE Home > Th. List > df-preset | Structured version Visualization version Unicode version |
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.) |
Ref | Expression |
---|---|
df-preset |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpreset 16926 | . 2 | |
2 | vx | . . . . . . . . . . 11 | |
3 | 2 | cv 1482 | . . . . . . . . . 10 |
4 | vr | . . . . . . . . . . 11 | |
5 | 4 | cv 1482 | . . . . . . . . . 10 |
6 | 3, 3, 5 | wbr 4653 | . . . . . . . . 9 |
7 | vy | . . . . . . . . . . . . 13 | |
8 | 7 | cv 1482 | . . . . . . . . . . . 12 |
9 | 3, 8, 5 | wbr 4653 | . . . . . . . . . . 11 |
10 | vz | . . . . . . . . . . . . 13 | |
11 | 10 | cv 1482 | . . . . . . . . . . . 12 |
12 | 8, 11, 5 | wbr 4653 | . . . . . . . . . . 11 |
13 | 9, 12 | wa 384 | . . . . . . . . . 10 |
14 | 3, 11, 5 | wbr 4653 | . . . . . . . . . 10 |
15 | 13, 14 | wi 4 | . . . . . . . . 9 |
16 | 6, 15 | wa 384 | . . . . . . . 8 |
17 | vb | . . . . . . . . 9 | |
18 | 17 | cv 1482 | . . . . . . . 8 |
19 | 16, 10, 18 | wral 2912 | . . . . . . 7 |
20 | 19, 7, 18 | wral 2912 | . . . . . 6 |
21 | 20, 2, 18 | wral 2912 | . . . . 5 |
22 | vf | . . . . . . 7 | |
23 | 22 | cv 1482 | . . . . . 6 |
24 | cple 15948 | . . . . . 6 | |
25 | 23, 24 | cfv 5888 | . . . . 5 |
26 | 21, 4, 25 | wsbc 3435 | . . . 4 |
27 | cbs 15857 | . . . . 5 | |
28 | 23, 27 | cfv 5888 | . . . 4 |
29 | 26, 17, 28 | wsbc 3435 | . . 3 |
30 | 29, 22 | cab 2608 | . 2 |
31 | 1, 30 | wceq 1483 | 1 |
Colors of variables: wff setvar class |
This definition is referenced by: isprs 16930 |
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