Theorem rankval2b

Description: Value of an alternate definition of the rank function. Definition of BellMachover p. 478. This variant of rankval2 does not use Regularity, and so requires the assumption that A is in the range of R1 . (Contributed by BTernaryTau, 19-Jan-2026)

	Ref	Expression
Assertion	rankval2b	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]\to \mathrm{rank}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|A\subseteq R_1\left(x\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		rankvalb	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]\to \mathrm{rank}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|A\in R_1\left(\mathrm{suc}x\right)\right\}$$
2		r1suc	$$⊢x\in \mathrm{On}\to R_1\left(\mathrm{suc}x\right)=𝒫R_1\left(x\right)$$
3	2	eleq2d	$$⊢x\in \mathrm{On}\to \left(A\in R_1\left(\mathrm{suc}x\right)\leftrightarrow A\in 𝒫R_1\left(x\right)\right)$$
4		fvex	$$⊢R_1\left(x\right)\in \mathrm{V}$$
5	4	elpw2	$$⊢A\in 𝒫R_1\left(x\right)\leftrightarrow A\subseteq R_1\left(x\right)$$
6	3 5	bitrdi	$$⊢x\in \mathrm{On}\to \left(A\in R_1\left(\mathrm{suc}x\right)\leftrightarrow A\subseteq R_1\left(x\right)\right)$$
7	6	rabbiia	$$⊢\left\{x\in \mathrm{On}|A\in R_1\left(\mathrm{suc}x\right)\right\}=\left\{x\in \mathrm{On}|A\subseteq R_1\left(x\right)\right\}$$
8	7	inteqi	$$⊢\bigcap \left\{x\in \mathrm{On}|A\in R_1\left(\mathrm{suc}x\right)\right\}=\bigcap \left\{x\in \mathrm{On}|A\subseteq R_1\left(x\right)\right\}$$
9	1 8	eqtrdi	$$⊢A\in \bigcup R_1\left[\mathrm{On}\right]\to \mathrm{rank}\left(A\right)=\bigcap \left\{x\in \mathrm{On}|A\subseteq R_1\left(x\right)\right\}$$