Metamath Proof Explorer


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 φFV
frege126d.x φXdomF
frege126d.a φA=FX
frege126d.xb φXt+FB
frege126d.fun φFunF
Assertion frege126d φAt+FBA=BBt+FA

Proof

Step Hyp Ref Expression
1 frege126d.f φFV
2 frege126d.x φXdomF
3 frege126d.a φA=FX
4 frege126d.xb φXt+FB
5 frege126d.fun φFunF
6 1 2 3 4 5 frege124d φAt+FBA=B
7 6 frege114d φAt+FBA=BBt+FA