View source for Equivalence classes are either equal or disjoint

{{Stub page|grade=B|msg=I'm sure I've already done this SOMEWHERE - find it!}}
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==Statement==
Let {{M|X}} be a [[set]], let {{M|\sim\subseteq X\times X}} be an [[equivalence relation]] on {{M|X}}, let {{M|\frac{X}{\sim} }} denote the [[quotient by an equivalence relation|quotient of {{M|X}} by {{M|\sim}}]]<ref group="Note">In other words: {{M|\frac{X}{\sim} }} is the set of [[equivalence classes]] of {{M|\sim}}</ref> and lastly let {{M|\pi:X\rightarrow\frac{X}{\sim} }} given by {{M|\pi:x\mapsto[x]}} be the [[canonical projection of the equivalence relation]]. Then: 
* We claim that {{M|\frac{X}{\sim} }} is a {{link|partition|abstract algebra}} of {{M|X}}. That is:
*# {{M|1=\forall x\in X\exists y\in\frac{X}{\sim}[x\in y]}} - all elements of {{M|x}} belong to an element of the partition
*# {{M|1=\forall u,v\in\frac{X}{\sim}[u\cap v\neq\emptyset\implies u\eq v]}} - if {{M|u}} and {{M|v}} are not [[disjoint]], they are equal
*#* Equivalently (by [[contrapositive]]<ref group="Note">The contrapositive of {{M|A\implies B}} is {{M|(\neg B)\implies(\neg A)}}. That is to say:
* {{M|\big(A\implies B\big)\iff\big((\neg B)\implies(\neg A)\big)}}</ref>): {{M|\forall u,v\in\frac{X}{\sim}[u\neq v\implies u\cap v\eq\emptyset]}}
==Proof==
{{Requires proof|grade=B|easy=yes|msg=This is an easy and routine proof. First year friendly and all. However:
* {{Caution|I must make sure to prove the requirements}} e.g. that {{M|[x]}}, the [[equivalence class]] containing {{M|x}} makes sense. As if I use a property like:
** {{M|y\in [x]\iff y\sim x}}
*: I'm sort of indirectly using part 2 of this theorem if I ever use the transitive property of equivalence relations involving {{M|x}} and {{M|y}}.
Basically - be careful. It's easy to be circular here as the results are so engrained into my every day mathematical work}}
==Notes==
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==References==
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{{Requires references|grade=D|msg={{Highlight|Do not worry I know this content really well, this result is true. I promise}}
* I need to find a book that deems this theorem worthy of making explicit!}}
{{Theorem Of|Abstract Algebra|Elementary Set Theory|Set Theory}}