Theorem smodm 8373

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 8368	. 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 1146	1 ⊢ (Smo 𝐴 → Ord dom 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2107  ∀wral 3050  dom cdm 5665  Ord word 6362  Oncon0 6363  ⟶wf 6537  ‘cfv 6541  Smo wsmo 8367

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 207  df-an 396  df-3an 1088  df-smo 8368

This theorem is referenced by:  smores2  8376  smodm2  8377  smoel  8382