We defined a graph property the following way:
Let $\Omega_n$ be the set of all graphs $G = ([n], E)$ on $n$ nodes. Then, a graph property is a sequence $P = (P_n)_{n \in \mathbb N}$ with $P_n \subseteq \Omega_n.$
Now, an example of a (especially monotone) graph property would be the connectivity of a graph.
At the same time, we introduced the Erdos-Renyi model $G(n,p)$, so given a graph with $n$ vertices, each edge exists with probability $p$ (or doesn't exist with probability $1-p$).
What confuses me is the another notation we introduced, namely
$$G(n,p) \in P.$$
What is that supposed to mean? Say, $P$ is the graph property of connectivity. We define $p = 1$, so given a set of $n$ vertices, every possible edge must exist. So, for example, if $n = 4$, the only graph following the model would be the complete graph $K_4$. But in this case, it doesn't make sense to write
$$G(4,1) \in P$$
since there are several other graphs with $4$ vertices that are connected, but $G(4,1)$ only consists of $K_4$. So it should be something like $G(4,1) \in P_4$, and $P_4$ is part of the sequence $P = (P_1, P_2, P_3, P_4, ...)$, but $G(4,1)$ clearly isn't an element of $P$ itself then.
It seems to me that this is just a very slight abuse of notation. Writing ${G(n,p) \in P}$ is supposed to mean that the random graph has the property described by $P$, i.e. it is a random event.
If one were to be extremely precise, $G(4,1) \in P_4$ is not quite right either, since $G(4,1)$ is not a graph, but a graph-valued random variable (though admittedly it is constant). On the other hand, it is standard notation in probability theory to write $\{X \in A\}$ instead of $\{\omega \in \Omega: X(\omega) \in A\}$ for a random variable $X : \Omega \rightarrow E$ and $A \subseteq E$, and your example is not too far off from this.