Theorem uniin

Description: The class union of the intersection of two classes. Exercise 4.12(n) of Mendelson p. 235. See uniinqs for a condition where equality holds. (Contributed by NM, 4-Dec-2003) (Proof shortened by Andrew Salmon, 29-Jun-2011) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026)

	Ref	Expression
Assertion	uniin	$$⊢\bigcup \left(A\cap B\right)\subseteq \bigcup A\cap \bigcup B$$

Proof

Step	Hyp	Ref	Expression
1		inss1	$$⊢A\cap B\subseteq A$$
2	1	unissi	$$⊢\bigcup \left(A\cap B\right)\subseteq \bigcup A$$
3		inss2	$$⊢A\cap B\subseteq B$$
4	3	unissi	$$⊢\bigcup \left(A\cap B\right)\subseteq \bigcup B$$
5	2 4	ssini	$$⊢\bigcup \left(A\cap B\right)\subseteq \bigcup A\cap \bigcup B$$