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 = { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) } |
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 | ⊢ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) |
11 | 10 2 0 | crab | ⊢ { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) } |
12 | 1 11 | wceq | ⊢ Scott 𝐴 = { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) } |