Theorem unielid

Description: Two ways to say the union of a class is an element of that class. (Contributed by RP, 27-Jan-2025)

	Ref	Expression
Assertion	unielid	$$⊢\bigcup A\in A\leftrightarrow \exists x\in A\forall y\in Ay\subseteq x$$

Proof

Step	Hyp	Ref	Expression
1		ssid	$$⊢A\subseteq A$$
2		unielss	$$⊢A\subseteq A\to \left(\bigcup A\in A\leftrightarrow \exists x\in A\forall y\in Ay\subseteq x\right)$$
3	1 2	ax-mp	$$⊢\bigcup A\in A\leftrightarrow \exists x\in A\forall y\in Ay\subseteq x$$