Theorem smodm 8397

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 8392	. 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 1164	1 ⊢ (Smo 𝐴 → Ord dom 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ∀wral 3080  dom cdm 5658  Ord word 6387  Oncon0 6388  ⟶wf 6562  ‘cfv 6566  Smo wsmo 8391

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-an 402  df-3an 1105  df-smo 8392

This theorem is used by:  smores2  8400  smodm2  8401  smoel  8406