Theorem expanduniss

Description: Expand U. A C_ B to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Assertion	expanduniss	$$⊢\bigcup A\subseteq B\leftrightarrow \forall x\left(x\in A\to \forall y\left(y\in x\to y\in B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		unissb	$$⊢\bigcup A\subseteq B\leftrightarrow \forall x\in Ax\subseteq B$$
2		df-ss	$$⊢x\subseteq B\leftrightarrow \forall y\left(y\in x\to y\in B\right)$$
3	2	expandral	$$⊢\forall x\in Ax\subseteq B\leftrightarrow \forall x\left(x\in A\to \forall y\left(y\in x\to y\in B\right)\right)$$
4	1 3	bitri	$$⊢\bigcup A\subseteq B\leftrightarrow \forall x\left(x\in A\to \forall y\left(y\in x\to y\in B\right)\right)$$