Theorem smodm 5906

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 5901	. 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 920	1 |- ( Smo A -> Ord dom A )

Syntax hints:   -> wi 4   e. wcel 1393   A.wral 2306   Ord word 4099   Oncon0 4100   dom cdm 4345   -->wf 4898   ` cfv 4902   Smo wsmo 5900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100

This theorem depends on definitions:  df-bi 110  df-3an 887  df-smo 5901

This theorem is referenced by:  smores2  5909  smodm2  5910  smoel  5915