Theorem onsupeqmax

Description: Condition when the supremum of a set of ordinals is the maximum element of that set. (Contributed by RP, 24-Jan-2025)

	Ref	Expression
Assertion	onsupeqmax	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∪ 𝐴 ∈ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		unielid	⊢ ( ∪ 𝐴 ∈ 𝐴 ↔ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 )
2	1	a1i	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ( ∪ 𝐴 ∈ 𝐴 ↔ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ) )
3	2	bicomd	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉 ) → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∪ 𝐴 ∈ 𝐴 ) )