Theorem smodm 6562

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

Assertion

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

Proof of Theorem smodm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-smo 6557	. 2 ⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
2	1	simp2bi 1044	1 ⊢ (Smo 𝐴 → Ord dom 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2209  ∀wral 2528  Ord word 4507  Oncon0 4508  dom cdm 4774  ⟶wf 5373  ‘cfv 5377  Smo wsmo 6556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This proof depends on definitions:  df-bi 117  df-3an 1011  df-smo 6557

This theorem is used by:  smores2  6565  smodm2  6566  smoel  6571