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Definition df-ps 17247
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 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}

Detailed syntax breakdown of Definition df-ps
StepHypRef Expression
1 cps 17245 . 2 class PosetRel
2 vr . . . . . 6 setvar 𝑟
32cv 1522 . . . . 5 class 𝑟
43wrel 5148 . . . 4 wff Rel 𝑟
53, 3ccom 5147 . . . . 5 class (𝑟𝑟)
65, 3wss 3607 . . . 4 wff (𝑟𝑟) ⊆ 𝑟
73ccnv 5142 . . . . . 6 class 𝑟
83, 7cin 3606 . . . . 5 class (𝑟𝑟)
9 cid 5052 . . . . . 6 class I
103cuni 4468 . . . . . . 7 class 𝑟
1110cuni 4468 . . . . . 6 class 𝑟
129, 11cres 5145 . . . . 5 class ( I ↾ 𝑟)
138, 12wceq 1523 . . . 4 wff (𝑟𝑟) = ( I ↾ 𝑟)
144, 6, 13w3a 1054 . . 3 wff (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))
1514, 2cab 2637 . 2 class {𝑟 ∣ (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}
161, 15wceq 1523 1 wff PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}
Colors of variables: wff setvar class
This definition is referenced by:  isps  17249
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