Theorem frege87d

Description: If the images of both { A } and U are subsets of U and C follows A in the transitive closure of R and B follows C in R , then B is an element of U . Similar to Proposition 87 of Frege1879 p. 66. Compare with frege87 . (Contributed by RP, 15-Jul-2020)

	Ref	Expression
Hypotheses	frege87d.r	|- ( ph -> R e. _V )
	frege87d.a	|- ( ph -> A e. _V )
	frege87d.b	|- ( ph -> B e. _V )
	frege87d.c	|- ( ph -> C e. _V )
	frege87d.ac	|- ( ph -> A ( t+ ` R ) C )
	frege87d.cb	|- ( ph -> C R B )
	frege87d.ss	|- ( ph -> ( R " { A } ) C_ U )
	frege87d.he	|- ( ph -> ( R " U ) C_ U )
Assertion	frege87d	|- ( ph -> B e. U )

Proof

Step	Hyp	Ref	Expression
1		frege87d.r	|- ( ph -> R e. _V )
2		frege87d.a	|- ( ph -> A e. _V )
3		frege87d.b	|- ( ph -> B e. _V )
4		frege87d.c	|- ( ph -> C e. _V )
5		frege87d.ac	|- ( ph -> A ( t+ ` R ) C )
6		frege87d.cb	|- ( ph -> C R B )
7		frege87d.ss	|- ( ph -> ( R " { A } ) C_ U )
8		frege87d.he	|- ( ph -> ( R " U ) C_ U )
9	1 2 3 4 5 6	frege96d	|- ( ph -> A ( t+ ` R ) B )
10	1 2 3 9 8 7	frege77d	|- ( ph -> B e. U )