Theorem limexissupab

Description: An ordinal which is a limit ordinal is equal to the supremum of the class of all its elements. Lemma 2.13 of Schloeder p. 5. (Contributed by RP, 27-Jan-2025)

	Ref	Expression
Assertion	limexissupab	|- ( ( Lim A /\ A e. V ) -> A = sup ( { x | x e. A } , On , _E ) )

Proof

Step	Hyp	Ref	Expression
1		limuni	|- ( Lim A -> A = U. A )
2	1	adantr	|- ( ( Lim A /\ A e. V ) -> A = U. A )
3		limord	|- ( Lim A -> Ord A )
4		ordsson	|- ( Ord A -> A C_ On )
5	3 4	syl	|- ( Lim A -> A C_ On )
6		onsupuni	|- ( ( A C_ On /\ A e. V ) -> sup ( A , On , _E ) = U. A )
7	5 6	sylan	|- ( ( Lim A /\ A e. V ) -> sup ( A , On , _E ) = U. A )
8		abid1	|- A = { x | x e. A }
9		supeq1	|- ( A = { x | x e. A } -> sup ( A , On , _E ) = sup ( { x | x e. A } , On , _E ) )
10	8 9	mp1i	|- ( ( Lim A /\ A e. V ) -> sup ( A , On , _E ) = sup ( { x | x e. A } , On , _E ) )
11	2 7 10	3eqtr2d	|- ( ( Lim A /\ A e. V ) -> A = sup ( { x | x e. A } , On , _E ) )