Theorem smodm 6188

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 6183	. 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 997	1 |- ( Smo A -> Ord dom A )

Syntax hints:   -> wi 4   e. wcel 1480   A.wral 2416   Ord word 4284   Oncon0 4285   dom cdm 4539   -->wf 5119   ` cfv 5123   Smo wsmo 6182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106

This theorem depends on definitions:  df-bi 116  df-3an 964  df-smo 6183

This theorem is referenced by:  smores2  6191  smodm2  6192  smoel  6197