Theorem rankvaln

Description: Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 , unless A is a proper class.) (Contributed by Mario Carneiro, 22-Mar-2013) (Revised by Mario Carneiro, 10-Sep-2013)

	Ref	Expression
Assertion	rankvaln	$$⊢\neg A\in \bigcup R_1\left[\mathrm{On}\right]\to \mathrm{rank}\left(A\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		rankf	$$⊢\mathrm{rank}:\bigcup R_1\left[\mathrm{On}\right]⟶\mathrm{On}$$
2	1	fdmi	$$⊢\mathrm{dom}\mathrm{rank}=\bigcup R_1\left[\mathrm{On}\right]$$
3	2	eleq2i	$$⊢A\in \mathrm{dom}\mathrm{rank}\leftrightarrow A\in \bigcup R_1\left[\mathrm{On}\right]$$
4		ndmfv	$$⊢\neg A\in \mathrm{dom}\mathrm{rank}\to \mathrm{rank}\left(A\right)=\varnothing$$
5	3 4	sylnbir	$$⊢\neg A\in \bigcup R_1\left[\mathrm{On}\right]\to \mathrm{rank}\left(A\right)=\varnothing$$