Theorem supxrcld

Description: The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypothesis	supxrcld.1	|- ( ph -> A C_ RR* )
Assertion	supxrcld	|- ( ph -> sup ( A , RR* , < ) e. RR* )

Proof

Step	Hyp	Ref	Expression
1		supxrcld.1	|- ( ph -> A C_ RR* )
2		supxrcl	|- ( A C_ RR* -> sup ( A , RR* , < ) e. RR* )
3	1 2	syl	|- ( ph -> sup ( A , RR* , < ) e. RR* )