Metamath Proof Explorer


Definition df-rpss

Description: Define a relation which corresponds to proper subsethood df-pss on sets. This allows to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss . (Contributed by Stefan O'Rear, 2-Nov-2014)

Ref Expression
Assertion df-rpss [⊊] = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 ⊊ 𝑦 }

Detailed syntax breakdown

Step Hyp Ref Expression
0 crpss ⊢ [⊊]
1 vx ⊢ 𝑥
2 vy ⊢ 𝑦
3 1 cv ⊢ 𝑥
4 2 cv ⊢ 𝑦
5 3 4 wpss ⊢ 𝑥 ⊊ 𝑦
6 5 1 2 copab ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 ⊊ 𝑦 }
7 0 6 wceq ⊢ [⊊] = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 ⊊ 𝑦 }