Theorem scotteld

Description: The Scott operation sends inhabited classes to inhabited sets. (Contributed by Rohan Ridenour, 11-Aug-2023)

	Ref	Expression
Hypothesis	scotteld.1	$$⊢\phi \to \exists xx\in A$$
Assertion	scotteld	$$⊢\phi \to \exists xx\in ScottA$$

Proof

Step	Hyp	Ref	Expression
1		scotteld.1	$$⊢\phi \to \exists xx\in A$$
2		n0	$$⊢A\ne \varnothing \leftrightarrow \exists xx\in A$$
3	1 2	sylibr	$$⊢\phi \to A\ne \varnothing$$
4	3	neneqd	$$⊢\phi \to \neg A=\varnothing$$
5		scott0	$$⊢A=\varnothing \leftrightarrow \left\{y\in A|\forall z\in A\mathrm{rank}\left(y\right)\subseteq \mathrm{rank}\left(z\right)\right\}=\varnothing$$
6		df-scott	$$⊢ScottA=\left\{y\in A|\forall z\in A\mathrm{rank}\left(y\right)\subseteq \mathrm{rank}\left(z\right)\right\}$$
7	6	eqeq1i	$$⊢ScottA=\varnothing \leftrightarrow \left\{y\in A|\forall z\in A\mathrm{rank}\left(y\right)\subseteq \mathrm{rank}\left(z\right)\right\}=\varnothing$$
8	5 7	bitr4i	$$⊢A=\varnothing \leftrightarrow ScottA=\varnothing$$
9	4 8	sylnib	$$⊢\phi \to \neg ScottA=\varnothing$$
10	9	neqned	$$⊢\phi \to ScottA\ne \varnothing$$
11		n0	$$⊢ScottA\ne \varnothing \leftrightarrow \exists xx\in ScottA$$
12	10 11	sylib	$$⊢\phi \to \exists xx\in ScottA$$