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	$$⊢A=\varnothing \leftrightarrow ScottA=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		scott0	$$⊢A=\varnothing \leftrightarrow \left\{x\in A|\forall y\in A\mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right\}=\varnothing$$
2		df-scott	$$⊢ScottA=\left\{x\in A|\forall y\in A\mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right\}$$
3	2	eqeq1i	$$⊢ScottA=\varnothing \leftrightarrow \left\{x\in A|\forall y\in A\mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right\}=\varnothing$$
4	1 3	bitr4i	$$⊢A=\varnothing \leftrightarrow ScottA=\varnothing$$