Metamath Proof Explorer


Definition df-scott

Description: Define the Scott operation. This operation constructs a subset of the input class which is nonempty whenever its input is using Scott's trick. (Contributed by Rohan Ridenour, 9-Aug-2023)

Ref Expression
Assertion df-scott Scott 𝐴 = { 𝑥𝐴 ∣ ∀ 𝑦𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA 𝐴
1 0 cscott Scott 𝐴
2 vx 𝑥
3 vy 𝑦
4 crnk rank
5 2 cv 𝑥
6 5 4 cfv ( rank ‘ 𝑥 )
7 3 cv 𝑦
8 7 4 cfv ( rank ‘ 𝑦 )
9 6 8 wss ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 )
10 9 3 0 wral 𝑦𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 )
11 10 2 0 crab { 𝑥𝐴 ∣ ∀ 𝑦𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) }
12 1 11 wceq Scott 𝐴 = { 𝑥𝐴 ∣ ∀ 𝑦𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) }