Theorem bnj1138

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj1138.1	$$⊢A=B\cup C$$
Assertion	bnj1138	$$⊢X\in A\leftrightarrow \left(X\in B\vee X\in C\right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj1138.1	$$⊢A=B\cup C$$
2	1	eleq2i	$$⊢X\in A\leftrightarrow X\in \left(B\cup C\right)$$
3		elun	$$⊢X\in \left(B\cup C\right)\leftrightarrow \left(X\in B\vee X\in C\right)$$
4	2 3	bitri	$$⊢X\in A\leftrightarrow \left(X\in B\vee X\in C\right)$$