In an attempt at proving a [proposition regarding power sets](https://math.stackexchange.com/questions/3270680/prove-if-pc-pa-cup-pb-then-a-c-or-b-c/3272591#3272591) I make use of the alleged theorem , that seems to me intuitively correct : 

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> If set $X$ is not equal to set $Y$, then $X\cup Z$ is not equal to $Y\cup Z$ ( for all $Z$).
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Is this proposition correct? 

In case it is, does it have a name?