Theorem frege86

Description: Conclusion about element one past Y in the R -sequence. Proposition 86 of Frege1879 p. 66. (Contributed by RP, 1-Jul-2020) (Revised by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege86.x	⊢ 𝑋 ∈ 𝑈
	frege86.y	⊢ 𝑌 ∈ 𝑉
	frege86.r	⊢ 𝑅 ∈ 𝑊
	frege86.a	⊢ 𝐴 ∈ 𝐵
Assertion	frege86	⊢ ( ( ( 𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑌 𝑅 𝑍 → 𝑍 ∈ 𝐴 ) ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑌 𝑅 𝑍 → 𝑍 ∈ 𝐴 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege86.x	⊢ 𝑋 ∈ 𝑈
2		frege86.y	⊢ 𝑌 ∈ 𝑉
3		frege86.r	⊢ 𝑅 ∈ 𝑊
4		frege86.a	⊢ 𝐴 ∈ 𝐵
5	1 2 3 4	frege85	⊢ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴 ) ) )
6		frege19	⊢ ( ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴 ) ) ) → ( ( ( 𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑌 𝑅 𝑍 → 𝑍 ∈ 𝐴 ) ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑌 𝑅 𝑍 → 𝑍 ∈ 𝐴 ) ) ) ) ) )
7	5 6	ax-mp	⊢ ( ( ( 𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑌 𝑅 𝑍 → 𝑍 ∈ 𝐴 ) ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ∀ 𝑤 ( 𝑋 𝑅 𝑤 → 𝑤 ∈ 𝐴 ) → ( 𝑅 hereditary 𝐴 → ( 𝑌 𝑅 𝑍 → 𝑍 ∈ 𝐴 ) ) ) ) )