Why every projective irreducible representation of $(ISO(2,1)^{\uparrow}=SO(2,1)^{\uparrow}\ltimes \mathbb{R}^3)$ is equivalent?

Question

Why every projective irreducible representation of the connected poincare group $(ISO(2,1)^{\uparrow}=SO(2,1)^{\uparrow}\ltimes \mathbb{R}^3)$ is equivalent to a projective irreducible representation of its double cover $(SL(2,\mathbb{R})\ltimes \mathfrak{sl}_{Ad}(2,\mathbb{R}))$? where $Ad:SL(2,\mathbb{R})\to \mathfrak{sl}_{Ad}(2,\mathbb{R})$ is the adjoint representation. I have also this question that why very projective irreducible representation of the connected Lie group is equivalent to an exact irreducible representation of its universal cover. Are these statements theorems? Unfortunately I don't understand the language of "The quantum theory of fields" book by S. Weinberg and it's a little bit hard to me. Is there a good reference for that?