Theorem smodm

Description: The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011)

	Ref	Expression
Assertion	smodm	⊢ ( Smo 𝐴 → Ord dom 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		df-smo	⊢ ( Smo 𝐴 ↔ ( 𝐴 : dom 𝐴 ⟶ On ∧ Ord dom 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∀ 𝑦 ∈ dom 𝐴 ( 𝑥 ∈ 𝑦 → ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐴 ‘ 𝑦 ) ) ) )
2	1	simp2bi	⊢ ( Smo 𝐴 → Ord dom 𝐴 )