$X$ and $Y$ are disjoint if
\[X\cap Y = \varnothing.\]$X$ is a disjoint collection if
\[\bigcap X = \varnothing.\]Not disjoint sets are called joint / overlapping.
$\varnothing$ is the only set disjoint to itself.
Proof.
- \[\varnothing\cap\varnothing = \varnothing.\]
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If $X$ is disjoint to itself,
\[X = X\cap X = \varnothing.\]
$\varnothing$ is the only set disjoint to any other set.
Proof.
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$\varnothing$ is disjoint to any set by IT#PROP-EMP (II).
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If $X$ is non-empty,
\[Y = X\cup\{X\}\]is always joint to $X$. By ZF#PROP-SLF, $X\neq Y$.