MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smodm Structured version   Visualization version   GIF version

Theorem smodm 7617
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 7612 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
21simp2bi 1141 1 (Smo 𝐴 → Ord dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  wral 3050  dom cdm 5266  Ord word 5883  Oncon0 5884  wf 6045  cfv 6049  Smo wsmo 7611
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 197  df-an 385  df-3an 1074  df-smo 7612
This theorem is referenced by:  smores2  7620  smodm2  7621  smoel  7626
  Copyright terms: Public domain W3C validator