Theorem elom

Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 . (Contributed by NM, 15-May-1994)

	Ref	Expression
Assertion	elom	|- ( A e. _om <-> ( A e. On /\ A. x ( Lim x -> A e. x ) ) )

Proof

Step	Hyp	Ref	Expression
1		elequ1	|- ( y = w -> ( y e. x <-> w e. x ) )
2	1	imbi2d	|- ( y = w -> ( ( Lim x -> y e. x ) <-> ( Lim x -> w e. x ) ) )
3	2	albidv	|- ( y = w -> ( A. x ( Lim x -> y e. x ) <-> A. x ( Lim x -> w e. x ) ) )
4		eleq1	|- ( w = A -> ( w e. x <-> A e. x ) )
5	4	imbi2d	|- ( w = A -> ( ( Lim x -> w e. x ) <-> ( Lim x -> A e. x ) ) )
6	5	albidv	|- ( w = A -> ( A. x ( Lim x -> w e. x ) <-> A. x ( Lim x -> A e. x ) ) )
7		df-om	|- _om = { y e. On | A. x ( Lim x -> y e. x ) }
8	3 6 7	elrab2w	|- ( A e. _om <-> ( A e. On /\ A. x ( Lim x -> A e. x ) ) )