Theorem smodm 6349

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 6344	. 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 1015	1 ⊢ (Smo 𝐴 → Ord dom 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2167  ∀wral 2475  Ord word 4397  Oncon0 4398  dom cdm 4663  ⟶wf 5254  ‘cfv 5258  Smo wsmo 6343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem depends on definitions:  df-bi 117  df-3an 982  df-smo 6344

This theorem is referenced by:  smores2  6352  smodm2  6353  smoel  6358