Theorem smodm 7448

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 7443	. 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 1077	1 |- ( Smo A -> Ord dom A )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   dom cdm 5114   Ord word 5722   Oncon0 5723   -->wf 5884   ` cfv 5888   Smo wsmo 7442

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 386  df-3an 1039  df-smo 7443

This theorem is referenced by:  smores2  7451  smodm2  7452  smoel  7457