Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:
$$x^2\ne x\implies x\ne 1$$
I immediately answered true, but for some reason, everyone (including my classmates and math teacher) is disagreeing with me. According to them, when $x^2$ is not equal to $x$, $x$ also can't be $0$ and because $0$ isn't excluded as a possible value of $x$, the sentence is false. After hours, I am still unable to understand this ridiculously simple implication. I can't believe I'm stuck with something so simple.
Why I think the logical sentence above is true:
My understanding of the implication symbol $\implies$ is the following:
If the left part is true, then the right part must be also true. If the left part is false, then nothing is said about the right part. In the right part of this specific implication nothing is said about whether $x$ can be $0$. Maybe $x$ can't be $-\pi i$ too, but as I see it, it doesn't really matter, as long as $x \ne 1$ holds. And it always holds when $x^2 \ne x$, therefore the sentence is true.
TL;DR:
$x^2 \ne x \implies x \ne 1$: Is this sentence true or false, and why?
Sorry for bothering such an amazing community with such a simple question, but I had to ask someone.

First, some general remarks about logical implications/conditional statements.   


As you know, $P \rightarrow Q$ is true when $P$ is false, or when $Q$ is true.   
As mentioned in the comments, the contrapositive of the implication $P \rightarrow Q$, written $\lnot Q \rightarrow \lnot P$, is logically equivalent to the implication.  
It is possible to write implications with merely the "or" operator. Namely, $P \rightarrow Q$ is equivalent to $\lnot P\text{ or }Q$, or in symbols, $\lnot P\lor Q$.


Now we can look at your specific case, using the above approaches. 


If $P$ is false, ie if $x^2 \neq x$ is false (so $x^2 = x$ ), then the statement is true, so we assume that $P$ is true. So, as a statement, $x^2 = x$ is false. Your teacher and classmates are rightly convinced that $x^2 = x$ is equivalent to ($x = 1$ or $x =0\;$), and we will use this here. 
If $P$ is true, then ($x=1\text{ or }x =0\;$) is false. In other words, ($x=1$) AND ($x=0\;$) are both false. I.e., ($x \neq 1$) and ($x \neq 0\;$) are true. 
I.e., if $P$, then $Q$. 
The contrapositive is $x = 1 \rightarrow x^2 = x$. True.
We use the "sufficiency of or" to write our conditional as: $$\lnot(x^2 \neq x)\lor x \neq 1\;.$$ That is, $x^2 = x$ or $x \neq 1$,
which is $$(x = 1\text{ or }x =0)\text{ or }x \neq 1,$$ which is
$$(x = 1\text{ or }x \neq 1)\text{ or }x = 0\;,$$ which is
$$(\text{TRUE})\text{ or }x = 0\;,$$ which is true. 

