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 ∈ 𝑈
	frege81.y	⊢ 𝑌 ∈ 𝑉
	frege81.r	⊢ 𝑅 ∈ 𝑊
	frege81.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege81	⊢ ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege81.x	⊢ 𝑋 ∈ 𝑈
2		frege81.y	⊢ 𝑌 ∈ 𝑉
3		frege81.r	⊢ 𝑅 ∈ 𝑊
4		frege81.a	⊢ 𝐴 ∈ 𝐵
5		vex	⊢ 𝑎 ∈ V
6	1 5	frege74	⊢ ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) ) )
7	6	alrimdv	⊢ ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) ) )
8	1 2 3 4	frege80	⊢ ( ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ∀ 𝑎 ( 𝑋 𝑅 𝑎 → 𝑎 ∈ 𝐴 ) ) ) → ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) ) )
9	7 8	ax-mp	⊢ ( 𝑋 ∈ 𝐴 → ( 𝑅 hereditary 𝐴 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → 𝑌 ∈ 𝐴 ) ) )