Metamath Proof Explorer

Theorem onsupintrab

Description: The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Definition 2.9 of Schloeder p. 5. (Contributed by RP, 23-Jan-2025)

		Ref	Expression
	Assertion	onsupintrab	$⊢ (A \subseteq On \land A \in V) \to sup (A, On, E) = ⋂ \{x \in On \| \forall y \in A y \subseteq x\}$

Proof

Step	Hyp	Ref	Expression
1		onsupuni	$⊢ (A \subseteq On \land A \in V) \to sup (A, On, E) = ⋃ A$
2		onuniintrab	$⊢ (A \subseteq On \land A \in V) \to ⋃ A = ⋂ \{x \in On \| \forall y \in A y \subseteq x\}$
3	1 2	eqtrd	$⊢ (A \subseteq On \land A \in V) \to sup (A, On, E) = ⋂ \{x \in On \| \forall y \in A y \subseteq x\}$