Theorem smodm 6181

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 6176	. 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 997	1 ⊢ (Smo 𝐴 → Ord dom 𝐴)

Syntax hints:   → wi 4   ∈ wcel 1480  ∀wral 2414  Ord word 4279  Oncon0 4280  dom cdm 4534  ⟶wf 5114  ‘cfv 5118  Smo wsmo 6175

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 964  df-smo 6176

This theorem is referenced by:  smores2  6184  smodm2  6185  smoel  6190