Theorem scott0bOLD

Description: Obsolete version of scott0b as of 18-Jul-2026. (Contributed by BTernaryTau, 3-Jul-2026) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	scott0bOLD	⊢ ( 𝐴 = ∅ ↔ Scott 𝐴 = ∅ )

Proof

Step	Hyp	Ref	Expression
1		scott0OLD	⊢ ( 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 = ∅ )