Theorem onsuplub

Description: The supremum of a set of ordinals is the least upper bound. (Contributed by RP, 27-Jan-2025)

	Ref	Expression
Assertion	onsuplub	⊢ ( ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) ∧ 𝐵 ∈ On ) → ( 𝐵 ∈ ∪ 𝐴 ↔ ∃ 𝑧 ∈ 𝐴 𝐵 ∈ 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		eluni2	⊢ ( 𝐵 ∈ ∪ 𝐴 ↔ ∃ 𝑧 ∈ 𝐴 𝐵 ∈ 𝑧 )
2	1	a1i	⊢ ( ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) ∧ 𝐵 ∈ On ) → ( 𝐵 ∈ ∪ 𝐴 ↔ ∃ 𝑧 ∈ 𝐴 𝐵 ∈ 𝑧 ) )