Theorem scott0b

Description: Applying Scott's trick yields the empty set iff it was applied to the empty set. (Contributed by BTernaryTau, 3-Jul-2026)

	Ref	Expression
Assertion	scott0b	⊢ ( 𝐴 = ∅ ↔ Scott 𝐴 = ∅ )

Proof

Step	Hyp	Ref	Expression
1		scott0	⊢ ( 𝐴 = ∅ ↔ { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) } = ∅ )
2		df-scott	⊢ Scott 𝐴 = { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) }
3	2	eqeq1i	⊢ ( Scott 𝐴 = ∅ ↔ { 𝑥 ∈ 𝐴 ∣ ∀ 𝑦 ∈ 𝐴 ( rank ‘ 𝑥 ) ⊆ ( rank ‘ 𝑦 ) } = ∅ )
4	1 3	bitr4i	⊢ ( 𝐴 = ∅ ↔ Scott 𝐴 = ∅ )