Theorem smodm 6259

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 6254	. 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 1003	1 ⊢ (Smo 𝐴 → Ord dom 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2136  ∀wral 2444  Ord word 4340  Oncon0 4341  dom cdm 4604  ⟶wf 5184  ‘cfv 5188  Smo wsmo 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106

This theorem depends on definitions:  df-bi 116  df-3an 970  df-smo 6254

This theorem is referenced by:  smores2  6262  smodm2  6263  smoel  6268