Theorem scottidm 35817

Description: The Scott operation is idempotent. (Contributed by BTernaryTau, 23-Aug-2026.)

Assertion

Ref	Expression
scottidm	⊢ Scott Scott 𝐴 = Scott 𝐴

Proof of Theorem scottidm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scottss 9942	. 2 ⊢ Scott Scott 𝐴 ⊆ Scott 𝐴
2		elscott 35801	. . . . . 6 ⊢ (𝑥 ∈ Scott 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)))
3		scottss 9942	. . . . . . 7 ⊢ Scott 𝐴 ⊆ 𝐴
4		ssralv 3978	. . . . . . 7 ⊢ (Scott 𝐴 ⊆ 𝐴 → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → ∀𝑦 ∈ Scott 𝐴(rank‘𝑥) ⊆ (rank‘𝑦)))
5	3, 4	ax-mp 5	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) → ∀𝑦 ∈ Scott 𝐴(rank‘𝑥) ⊆ (rank‘𝑦))
6	2, 5	simplbiim 514	. . . . 5 ⊢ (𝑥 ∈ Scott 𝐴 → ∀𝑦 ∈ Scott 𝐴(rank‘𝑥) ⊆ (rank‘𝑦))
7	6	ancli 558	. . . 4 ⊢ (𝑥 ∈ Scott 𝐴 → (𝑥 ∈ Scott 𝐴 ∧ ∀𝑦 ∈ Scott 𝐴(rank‘𝑥) ⊆ (rank‘𝑦)))
8		elscott 35801	. . . 4 ⊢ (𝑥 ∈ Scott Scott 𝐴 ↔ (𝑥 ∈ Scott 𝐴 ∧ ∀𝑦 ∈ Scott 𝐴(rank‘𝑥) ⊆ (rank‘𝑦)))
9	7, 8	sylibr 237	. . 3 ⊢ (𝑥 ∈ Scott 𝐴 → 𝑥 ∈ Scott Scott 𝐴)
10	9	ssriv 3913	. 2 ⊢ Scott 𝐴 ⊆ Scott Scott 𝐴
11	1, 10	eqssi 3925	1 ⊢ Scott Scott 𝐴 = Scott 𝐴

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3896  ‘cfv 6543  rankcrnk 9774  Scott cscott 9935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-scott 9936

This theorem is used by: (None)