Definition df-scott 44473

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.)

Assertion

Ref	Expression
df-scott	⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}

Distinct variable group:   𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-scott

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	cscott 44472	. 2 class Scott 𝐴
3		vx	. . . . . . 7 setvar 𝑥
4	3	cv 1540	. . . . . 6 class 𝑥
5		crnk 9675	. . . . . 6 class rank
6	4, 5	cfv 6492	. . . . 5 class (rank‘𝑥)
7		vy	. . . . . . 7 setvar 𝑦
8	7	cv 1540	. . . . . 6 class 𝑦
9	8, 5	cfv 6492	. . . . 5 class (rank‘𝑦)
10	6, 9	wss 3901	. . . 4 wff (rank‘𝑥) ⊆ (rank‘𝑦)
11	10, 7, 1	wral 3051	. . 3 wff ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)
12	11, 3, 1	crab 3399	. 2 class {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
13	2, 12	wceq 1541	1 wff Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}

This definition is referenced by:  scotteqd  44474  nfscott  44476  scottabf  44477  scottss  44480  scottex2  44482  scotteld  44483  scottelrankd  44484