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Theorem smodm 6232
 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.
StepHypRef Expression
1 df-smo 6227 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
21simp2bi 998 1 (Smo 𝐴 → Ord dom 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 2128  ∀wral 2435  Ord word 4321  Oncon0 4322  dom cdm 4583  ⟶wf 5163  ‘cfv 5167  Smo wsmo 6226 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 965  df-smo 6227 This theorem is referenced by:  smores2  6235  smodm2  6236  smoel  6241
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