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	$$⊢\left(\left(A\subseteq \mathrm{On}\wedge A\in V\right)\wedge B\in \mathrm{On}\right)\to \left(B\in \bigcup A\leftrightarrow \exists z\in AB\in z\right)$$

Proof

Step	Hyp	Ref	Expression
1		eluni2	$$⊢B\in \bigcup A\leftrightarrow \exists z\in AB\in z$$
2	1	a1i	$$⊢\left(\left(A\subseteq \mathrm{On}\wedge A\in V\right)\wedge B\in \mathrm{On}\right)\to \left(B\in \bigcup A\leftrightarrow \exists z\in AB\in z\right)$$