Theorem smodm 6377

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 6372	. 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 1016	1 |- ( Smo A -> Ord dom A )

Syntax hints:   -> wi 4   e. wcel 2176   A.wral 2484   Ord word 4409   Oncon0 4410   dom cdm 4675   -->wf 5267   ` cfv 5271   Smo wsmo 6371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem depends on definitions:  df-bi 117  df-3an 983  df-smo 6372

This theorem is referenced by:  smores2  6380  smodm2  6381  smoel  6386