ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  smodm Unicode version

Theorem smodm 5940
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 5935 . 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 955 1  |-  ( Smo 
A  ->  Ord  dom  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   A.wral 2349   Ord word 4125   Oncon0 4126   dom cdm 4371   -->wf 4928   ` cfv 4932   Smo wsmo 5934
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 5935
This theorem is referenced by:  smores2  5943  smodm2  5944  smoel  5949
  Copyright terms: Public domain W3C validator