If $X$ is a disjoint collection, the union of $X$ is called a disjoint union. The notation
\[A = \bigsqcup X\]means
\[A = \bigcup X \enspace\land\enspace \bigcap X = \varnothing.\]We write
\[\begin{align*} X\sqcup Y &= \bigsqcup\{X,Y\}, \\ X\sqcup Y\sqcup Z &= (X\sqcup Y)\sqcup Z, \\ X\sqcup Y\sqcup Z\sqcup U &= (X\sqcup Y\sqcup Z)\sqcup U \end{align*}\]and so on.
If $X$ is a finite collection of disjoint sets,
\[\Big\lvert\bigsqcup X\Big\rvert = \sum_{A\in X}\lvert A\rvert.\]
Proof.By CDAR#PROP-ADD.