Theorem onsupuni2

Description: The supremum of a set of ordinals is the union of that set. (Contributed by RP, 22-Jan-2025)

	Ref	Expression
Assertion	onsupuni2	⊢ ( 𝐴 ∈ 𝒫 On → sup ( 𝐴 , On , E ) = ∪ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		elpwb	⊢ ( 𝐴 ∈ 𝒫 On ↔ ( 𝐴 ∈ V ∧ 𝐴 ⊆ On ) )
2		onsupuni	⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ∈ V ) → sup ( 𝐴 , On , E ) = ∪ 𝐴 )
3	2	ancoms	⊢ ( ( 𝐴 ∈ V ∧ 𝐴 ⊆ On ) → sup ( 𝐴 , On , E ) = ∪ 𝐴 )
4	1 3	sylbi	⊢ ( 𝐴 ∈ 𝒫 On → sup ( 𝐴 , On , E ) = ∪ 𝐴 )