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Theorem smodm 8372
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 8367 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
21simp2bi 1143 1 (Smo 𝐴 → Ord dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wral 3050  dom cdm 5678  Ord word 6370  Oncon0 6371  wf 6545  cfv 6549  Smo wsmo 8366
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 206  df-an 395  df-3an 1086  df-smo 8367
This theorem is referenced by:  smores2  8375  smodm2  8376  smoel  8381
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