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Theorem smodm 5960
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 5955 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
21simp2bi 955 1 (Smo 𝐴 → Ord dom 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  wral 2353  Ord word 4145  Oncon0 4146  dom cdm 4391  wf 4948  cfv 4952  Smo wsmo 5954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105
This theorem depends on definitions:  df-bi 115  df-3an 922  df-smo 5955
This theorem is referenced by:  smores2  5963  smodm2  5964  smoel  5969
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