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Definition df-ps 17408
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 17406 . 2 class PosetRel
2 vr . . . . . 6 setvar 𝑟
32cv 1636 . . . . 5 class 𝑟
43wrel 5323 . . . 4 wff Rel 𝑟
53, 3ccom 5322 . . . . 5 class (𝑟𝑟)
65, 3wss 3776 . . . 4 wff (𝑟𝑟) ⊆ 𝑟
73ccnv 5317 . . . . . 6 class 𝑟
83, 7cin 3775 . . . . 5 class (𝑟𝑟)
9 cid 5225 . . . . . 6 class I
103cuni 4637 . . . . . . 7 class 𝑟
1110cuni 4637 . . . . . 6 class 𝑟
129, 11cres 5320 . . . . 5 class ( I ↾ 𝑟)
138, 12wceq 1637 . . . 4 wff (𝑟𝑟) = ( I ↾ 𝑟)
144, 6, 13w3a 1100 . . 3 wff (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))
1514, 2cab 2799 . 2 class {𝑟 ∣ (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}
161, 15wceq 1637 1 wff PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}
Colors of variables: wff setvar class
This definition is referenced by:  isps  17410
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