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		eleq1	|- ( y = A -> ( y e. x <-> A e. x ) )
2	1	imbi2d	|- ( y = A -> ( ( Lim x -> y e. x ) <-> ( Lim x -> A e. x ) ) )
3	2	albidv	|- ( y = A -> ( A. x ( Lim x -> y e. x ) <-> A. x ( Lim x -> A e. x ) ) )
4		df-om	|- _om = { y e. On | A. x ( Lim x -> y e. x ) }
5	3 4	elrab2	|- ( A e. _om <-> ( A e. On /\ A. x ( Lim x -> A e. x ) ) )