Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-scott | Structured version Visualization version GIF version |
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 |
---|---|
df-scott | ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | 1 | cscott 41823 | . 2 class Scott 𝐴 |
3 | vx | . . . . . . 7 setvar 𝑥 | |
4 | 3 | cv 1541 | . . . . . 6 class 𝑥 |
5 | crnk 9522 | . . . . . 6 class rank | |
6 | 4, 5 | cfv 6432 | . . . . 5 class (rank‘𝑥) |
7 | vy | . . . . . . 7 setvar 𝑦 | |
8 | 7 | cv 1541 | . . . . . 6 class 𝑦 |
9 | 8, 5 | cfv 6432 | . . . . 5 class (rank‘𝑦) |
10 | 6, 9 | wss 3892 | . . . 4 wff (rank‘𝑥) ⊆ (rank‘𝑦) |
11 | 10, 7, 1 | wral 3066 | . . 3 wff ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) |
12 | 11, 3, 1 | crab 3070 | . 2 class {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} |
13 | 2, 12 | wceq 1542 | 1 wff Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} |
Colors of variables: wff setvar class |
This definition is referenced by: scotteqd 41825 nfscott 41827 scottabf 41828 scottss 41831 scottex2 41833 scotteld 41834 scottelrankd 41835 |
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