Theorem frege126d

Description: If F is a function, A is the successor of X , and B follows X in the transitive closure of F , then (for distinct A and B ) either A follows B or B follows A in the transitive closure of F . Similar to Proposition 126 of Frege1879 p. 81. Compare with frege126 . (Contributed by RP, 16-Jul-2020)

	Ref	Expression
Hypotheses	frege126d.f	|- ( ph -> F e. _V )
	frege126d.x	|- ( ph -> X e. dom F )
	frege126d.a	|- ( ph -> A = ( F ` X ) )
	frege126d.xb	|- ( ph -> X ( t+ ` F ) B )
	frege126d.fun	|- ( ph -> Fun F )
Assertion	frege126d	|- ( ph -> ( A ( t+ ` F ) B \/ A = B \/ B ( t+ ` F ) A ) )

Proof

Step	Hyp	Ref	Expression
1		frege126d.f	|- ( ph -> F e. _V )
2		frege126d.x	|- ( ph -> X e. dom F )
3		frege126d.a	|- ( ph -> A = ( F ` X ) )
4		frege126d.xb	|- ( ph -> X ( t+ ` F ) B )
5		frege126d.fun	|- ( ph -> Fun F )
6	1 2 3 4 5	frege124d	|- ( ph -> ( A ( t+ ` F ) B \/ A = B ) )
7	6	frege114d	|- ( ph -> ( A ( t+ ` F ) B \/ A = B \/ B ( t+ ` F ) A ) )