Theorem frege106d 41363

Description: If 𝐵 follows 𝐴 in 𝑅, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in 𝑅. Similar to Proposition 106 of [Frege1879] p. 73. Compare with frege106 41577. (Contributed by RP, 15-Jul-2020.)

Hypothesis

Ref	Expression
frege106d.cb	⊢ (𝜑 → 𝐴𝑅𝐵)

Assertion

Ref	Expression
frege106d	⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))

Proof of Theorem frege106d

Step	Hyp	Ref	Expression
1		frege106d.cb	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2	1	orcd 870	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∨ wo 844   = wceq 1539   class class class wbr 5074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-or 845

This theorem is referenced by:  frege108d  41364