Theorem winaon

Description: A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014)

	Ref	Expression
Assertion	winaon	|- ( A e. InaccW -> A e. On )

Proof

Step	Hyp	Ref	Expression
1		elwina	|- ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) )
2		cfon	|- ( cf ` A ) e. On
3		eleq1	|- ( ( cf ` A ) = A -> ( ( cf ` A ) e. On <-> A e. On ) )
4	2 3	mpbii	|- ( ( cf ` A ) = A -> A e. On )
5	4	3ad2ant2	|- ( ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) -> A e. On )
6	1 5	sylbi	|- ( A e. InaccW -> A e. On )