Theorem smodm 8351

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 8346	. 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 1147	1 ⊢ (Smo 𝐴 → Ord dom 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2107  ∀wral 3062  dom cdm 5677  Ord word 6364  Oncon0 6365  ⟶wf 6540  ‘cfv 6544  Smo wsmo 8345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-smo 8346

This theorem is referenced by:  smores2  8354  smodm2  8355  smoel  8360