Theorem alephsson

Description: The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers. (Contributed by NM, 11-Nov-2003)

	Ref	Expression
Assertion	alephsson	|- ran aleph C_ On

Proof

Step	Hyp	Ref	Expression
1		isinfcard	|- ( ( _om C_ x /\ ( card ` x ) = x ) <-> x e. ran aleph )
2		cardon	|- ( card ` x ) e. On
3		eleq1	|- ( ( card ` x ) = x -> ( ( card ` x ) e. On <-> x e. On ) )
4	2 3	mpbii	|- ( ( card ` x ) = x -> x e. On )
5	4	adantl	|- ( ( _om C_ x /\ ( card ` x ) = x ) -> x e. On )
6	1 5	sylbir	|- ( x e. ran aleph -> x e. On )
7	6	ssriv	|- ran aleph C_ On