Theorem dflim2

Description: An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004)

	Ref	Expression
Assertion	dflim2	|- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )

Proof

Step	Hyp	Ref	Expression
1		df-lim	|- ( Lim A <-> ( Ord A /\ A =/= (/) /\ A = U. A ) )
2		ord0eln0	|- ( Ord A -> ( (/) e. A <-> A =/= (/) ) )
3	2	anbi1d	|- ( Ord A -> ( ( (/) e. A /\ A = U. A ) <-> ( A =/= (/) /\ A = U. A ) ) )
4	3	pm5.32i	|- ( ( Ord A /\ ( (/) e. A /\ A = U. A ) ) <-> ( Ord A /\ ( A =/= (/) /\ A = U. A ) ) )
5		3anass	|- ( ( Ord A /\ (/) e. A /\ A = U. A ) <-> ( Ord A /\ ( (/) e. A /\ A = U. A ) ) )
6		3anass	|- ( ( Ord A /\ A =/= (/) /\ A = U. A ) <-> ( Ord A /\ ( A =/= (/) /\ A = U. A ) ) )
7	4 5 6	3bitr4i	|- ( ( Ord A /\ (/) e. A /\ A = U. A ) <-> ( Ord A /\ A =/= (/) /\ A = U. A ) )
8	1 7	bitr4i	|- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) )