Theorem undisj1

Description: The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004)

	Ref	Expression
Assertion	undisj1	$$⊢\left(A\cap C=\varnothing \wedge B\cap C=\varnothing \right)\leftrightarrow \left(A\cup B\right)\cap C=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		un00	$$⊢\left(A\cap C=\varnothing \wedge B\cap C=\varnothing \right)\leftrightarrow \left(A\cap C\right)\cup \left(B\cap C\right)=\varnothing$$
2		indir	$$⊢\left(A\cup B\right)\cap C=\left(A\cap C\right)\cup \left(B\cap C\right)$$
3	2	eqeq1i	$$⊢\left(A\cup B\right)\cap C=\varnothing \leftrightarrow \left(A\cap C\right)\cup \left(B\cap C\right)=\varnothing$$
4	1 3	bitr4i	$$⊢\left(A\cap C=\varnothing \wedge B\cap C=\varnothing \right)\leftrightarrow \left(A\cup B\right)\cap C=\varnothing$$