Theorem frege114

Description: If X belongs to the R -sequence beginning with Z , then Z belongs to the R -sequence beginning with X or X follows Z in the R -sequence. Proposition 114 of Frege1879 p. 76. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege114.x	⊢ 𝑋 ∈ 𝑈
	frege114.z	⊢ 𝑍 ∈ 𝑉
Assertion	frege114	⊢ ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 → ( ¬ 𝑍 ( t+ ‘ 𝑅 ) 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		frege114.x	⊢ 𝑋 ∈ 𝑈
2		frege114.z	⊢ 𝑍 ∈ 𝑉
3	1	frege104	⊢ ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 → ( ¬ 𝑍 ( t+ ‘ 𝑅 ) 𝑋 → 𝑍 = 𝑋 ) )
4	2	frege113	⊢ ( ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 → ( ¬ 𝑍 ( t+ ‘ 𝑅 ) 𝑋 → 𝑍 = 𝑋 ) ) → ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 → ( ¬ 𝑍 ( t+ ‘ 𝑅 ) 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) ) )
5	3 4	ax-mp	⊢ ( 𝑍 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑋 → ( ¬ 𝑍 ( t+ ‘ 𝑅 ) 𝑋 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑍 ) )