Theorem smodm 5929

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.

Step	Hyp	Ref	Expression
1		df-smo 5924	. 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 ) ) ) )
2	1	simp2bi 954	1 |- ( Smo A -> Ord dom A )

Syntax hints:   -> wi 4   e. wcel 1433   A.wral 2348   Ord word 4117   Oncon0 4118   dom cdm 4363   -->wf 4918   ` cfv 4922   Smo wsmo 5923

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 921  df-smo 5924

This theorem is referenced by:  smores2  5932  smodm2  5933  smoel  5938