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Theorem smodm 6613
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 6608 . 2  |-  ( Smo 
A  <->  ( A : dom  A --> On  /\  Ord  dom 
A  /\  A. x  e.  dom  A A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) ) ) )
21simp2bi 973 1  |-  ( Smo 
A  ->  Ord  dom  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   A.wral 2705   Ord word 4580   Oncon0 4581   dom cdm 4878   -->wf 5450   ` cfv 5454   Smo wsmo 6607
This theorem is referenced by:  smores2  6616  smodm2  6617  smoel  6622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-smo 6608
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