Theorem dfscott2

Description: Alternate definition of a Scott's trick set. (Contributed by BTernaryTau, 8-Jul-2026)

	Ref	Expression
Assertion	dfscott2	⊢ Scott 𝐴 = { 𝑥 ∈ 𝐴 ∣ ( rank ‘ 𝑥 ) = ∩ ( rank “ 𝐴 ) }

Proof

Step	Hyp	Ref	Expression
1		df-scott	⊢ Scott 𝐴 = { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) }
2		rankfn	⊢ rank Fn V
3		ssv	⊢ 𝐴 ⊆ V
4		fnfvintima	⊢ ( ( rank Fn V ∧ 𝐴 ⊆ V ∧ 𝑥 ∈ 𝐴 ) → ( ( rank ‘ 𝑥 ) = ∩ ( rank “ 𝐴 ) ↔ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) ) )
5	2 3 4	mp3an12	⊢ ( 𝑥 ∈ 𝐴 → ( ( rank ‘ 𝑥 ) = ∩ ( rank “ 𝐴 ) ↔ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) ) )
6	5	rabbiia	⊢ { 𝑥 ∈ 𝐴 ∣ ( rank ‘ 𝑥 ) = ∩ ( rank “ 𝐴 ) } = { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) }
7	1 6	eqtr4i	⊢ Scott 𝐴 = { 𝑥 ∈ 𝐴 ∣ ( rank ‘ 𝑥 ) = ∩ ( rank “ 𝐴 ) }