Symbol for 'implies and is not implied by'? Is there a mathematical symbol for 'implies and is not implied by'?
ie.
$$\Rightarrow \land \nLeftarrow$$
Context:
To provide extra emphasis in proofs with lines such as
$$x = 1 \,\,\,\,\,(\Rightarrow \land \nLeftarrow)\,\,\,\,\, x^2 = 1$$
 A: flawr has created one way to say what you're trying to say, but it isn't any better than your suggestion, in that neither will be readily understood unless you pre-define what the notation means.
To be understood clearly, simply write $$(A \to B) \land \lnot (B\to A)\tag{Use for greatest clarity (1)}$$ 
No need to create fancy notation for $(1)$, and in doing so, it will likely confuse otheres as to  what you're trying to say.
Edit
In response to the helpful comment from @T.Gunn, below:
If we take the question as a universal statement, with predicate "=", we can express the asker's $\Rightarrow$ as follows   $$\forall x ((x=1) \to (x^2 = 1))$$ whereas, we have $\not\Leftarrow$ means: $$\lnot \forall x((x^2=1)\to (x=1)) \equiv \exists x ((x^2 =1) \land (x\neq 1))\tag{$\dagger$}$$ 
$\dagger$: In this case, the existence of $x=-1$ verifies that $$\exists x (x^2 =1) \land (x\neq 1)$$
Given this universal interpretation, the notation used by the asker is not appropriate, as it might be for a propositional implication.  What we can say, is $$\forall x ((x=1) \to (x^2 = 1)) \land \lnot \forall x((x^2=1)\to (x=1))$$
A: I don't think there is an own symbol, but to express $A  \implies B$ and $B  \,\,\,\,\,\,\not\!\!\!\!\!\implies A$ I've seen
$\renewcommand{\arraystretch}{0.1}$
$$A \begin{array}{c} \Rightarrow \\ \nLeftarrow \end{array} B$$
