Theorem supxrcl

Description: The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005)

	Ref	Expression
Assertion	supxrcl	⊢ ( 𝐴 ⊆ ℝ* → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		xrltso	⊢ < Or ℝ*
2	1	a1i	⊢ ( 𝐴 ⊆ ℝ* → < Or ℝ* )
3		xrsupss	⊢ ( 𝐴 ⊆ ℝ* → ∃ 𝑥 ∈ ℝ* ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) )
4	2 3	supcl	⊢ ( 𝐴 ⊆ ℝ* → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* )