Theorem frege132

Description: Lemma for frege133 . Proposition 132 of Frege1879 p. 86. (Contributed by RP, 9-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege130.m	⊢ 𝑀 ∈ 𝑈
	frege130.r	⊢ 𝑅 ∈ 𝑉
Assertion	frege132	⊢ ( ( 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege130.m	⊢ 𝑀 ∈ 𝑈
2		frege130.r	⊢ 𝑅 ∈ 𝑉
3	1 2	frege131	⊢ ( Fun ◡ ◡ 𝑅 → 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) )
4		frege9	⊢ ( ( Fun ◡ ◡ 𝑅 → 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) → ( ( 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) ) )
5	3 4	ax-mp	⊢ ( ( 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) )