Theorem onf1onf1or 36051

Description: If 𝐹 maps 𝐴 one-to-one into the ordinals, then either 𝐴 can be mapped one-to-one into the ordinals (by 𝐹), or the ordinals can be mapped one-to-one into 𝐴. This is the ZFC version of (6 → 9) in https://tinyurl.com/hamkins-gblac. The proof is trivial but is included for completeness. (Contributed by BTernaryTau, 16-Sep-2026.)

Assertion

Ref	Expression
onf1onf1or	⊢ (𝐹:𝐴⟶1-1On → (𝐹:𝐴⟶1-1On ∨ 𝐻:On⟶1-1𝐴))

Proof of Theorem onf1onf1or

Step	Hyp	Ref	Expression
1		orc 881	1 ⊢ (𝐹:𝐴⟶1-1On → (𝐹:𝐴⟶1-1On ∨ 𝐻:On⟶1-1𝐴))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861  Oncon0 6380  ⟶1-1wf1 6555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862

This theorem is used by: (None)