Theorem frege122d 44546

Description: If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 is the successor of 𝑋, then 𝐴 and 𝐵 are the same (or 𝐵 follows 𝐴 in the transitive closure of 𝐹). Similar to Proposition 122 of [Frege1879] p. 79. Compare with frege122 44771. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege122d.a	⊢ (𝜑 → 𝐴 = (𝐹‘𝑋))
frege122d.b	⊢ (𝜑 → 𝐵 = (𝐹‘𝑋))

Assertion

Ref	Expression
frege122d	⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵))

Proof of Theorem frege122d

Step	Hyp	Ref	Expression
1		frege122d.a	. . 3 ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋))
2		frege122d.b	. . 3 ⊢ (𝜑 → 𝐵 = (𝐹‘𝑋))
3	1, 2	eqtr4d 2803	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	olcd 888	1 ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   class class class wbr 5111  ‘cfv 6540  t+ctcl 15048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-cleq 2757

This theorem is used by: (None)