Clarification regarding an exercise about primitive permutation group

Question

Let $G$ be a transitive group of permutations of a set $X$ and $H$ the stabiliser of an element $x\in X$. Show the equivalence of the following properties:

1. Every subgroup of $H'$ of $G$ containing $H$ is equal to $H$ or $G$.
2. Every subset $Y$ of $X$ such that, for all $g\in G$, $gY$ is either contained in $Y$ or disjoint from $Y$, is equal to $X$ or consists of a single element.

Presumably this means that $G\leq\mathfrak{S}_{X}$ ($\mathfrak{S}_{X}$ is the symmetric group of $X$) and that there exists an element $a\in X$ such that $$\omega_a(G)=X,$$ where $\omega_a$ is the orbital mapping defined by $a$. I don't quite understand what the exact equivalence it is that I am asked to prove. Is it an equivalence between the following statements?

3. $(\forall x)(x\in X\implies(\forall H')(H'\leq G\land\text{stab}(x)\subset H'\implies H'=\text{stab}(x)\lor H'=G))$;
4. $(\forall Y)(Y\subset X\land(\forall g)(g\in G\implies gY\subset Y\lor gY\cap Y=\emptyset)\implies Y=X\lor (\exists y)(Y=\{y\}))$.

Are $3,4$ correct translations of $1,2$?

Answer 1

I think that 3 and 4 are correct translations of 1. and 2. respectively. For your readers' sake at least, maybe you could write the set-element relations inside the defining quantifiers, when it is relevant.

I mean, write $(\forall x \in X) (x=x)$ (or $(\exists Y \subset X)(Y=Y)$) instead of $(\forall x)(x \in X \Rightarrow x=x)$ (or $(\exists Y)(Y \subset X \wedge Y=Y)$), as the first options are easier to understand.

Answer 2

$1. \;\text{stab}(x)<H\leq G\implies H=G$

$2.\;\;\; \forall Y\subset X\;[\forall g\in G\;gY\subseteq Y\lor gY\subseteq G-Y]\implies |Y|=1$