Theorem frege81

Description: If X has a property A that is hereditary in the R -sequence, and if Y follows X in the R -sequence, then Y has property A . This is a form of induction attributed to Jakob Bernoulli. Proposition 81 of Frege1879 p. 63. (Contributed by RP, 1-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege81.x	$$⊢X\in U$$
	frege81.y	$$⊢Y\in V$$
	frege81.r	$$⊢R\in W$$
	frege81.a	$$⊢A\in B$$
Assertion	frege81	$$⊢X\in A\to \left(RhereditaryA\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege81.x	$$⊢X\in U$$
2		frege81.y	$$⊢Y\in V$$
3		frege81.r	$$⊢R\in W$$
4		frege81.a	$$⊢A\in B$$
5		vex	$$⊢a\in \mathrm{V}$$
6	1 5	frege74	$$⊢X\in A\to \left(RhereditaryA\to \left(XRa\to a\in A\right)\right)$$
7	6	alrimdv	$$⊢X\in A\to \left(RhereditaryA\to \forall a\left(XRa\to a\in A\right)\right)$$
8	1 2 3 4	frege80	$$⊢\left(X\in A\to \left(RhereditaryA\to \forall a\left(XRa\to a\in A\right)\right)\right)\to \left(X\in A\to \left(RhereditaryA\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)\right)$$
9	7 8	ax-mp	$$⊢X\in A\to \left(RhereditaryA\to \left(X\mathrm{t+}\left(R\right)Y\to Y\in A\right)\right)$$