Theorem smodm 6566

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 6561	. 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 1044	1 ⊢ (Smo 𝐴 → Ord dom 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2209  ∀wral 2528  Ord word 4509  Oncon0 4510  dom cdm 4776  ⟶wf 5376  ‘cfv 5380  Smo wsmo 6560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This proof depends on definitions:  df-bi 117  df-3an 1011  df-smo 6561

This theorem is used by:  smores2  6569  smodm2  6570  smoel  6575