Theorem frege111

Description: If Y belongs to the R -sequence beginning with Z , then every result of an application of the procedure R to Y belongs to the R -sequence beginning with Z or precedes Z in the R -sequence. Proposition 111 of Frege1879 p. 75. (Contributed by RP, 7-Jul-2020) (Revised by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege111.z	$$⊢Z\in A$$
	frege111.y	$$⊢Y\in B$$
	frege111.v	$$⊢V\in C$$
	frege111.r	$$⊢R\in D$$
Assertion	frege111	$$⊢Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to \left(\neg V\mathrm{t+}\left(R\right)Z\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege111.z	$$⊢Z\in A$$
2		frege111.y	$$⊢Y\in B$$
3		frege111.v	$$⊢V\in C$$
4		frege111.r	$$⊢R\in D$$
5	1 2 3 4	frege108	$$⊢Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)$$
6		frege25	$$⊢\left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)\right)\to \left(Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to \left(\neg V\mathrm{t+}\left(R\right)Z\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)\right)\right)$$
7	5 6	ax-mp	$$⊢Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRV\to \left(\neg V\mathrm{t+}\left(R\right)Z\to Z\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)V\right)\right)$$