How do I improve the formulation of this condition?

This might look complicated, but it is not so much.

My idea is to have narrower width so the condition will stand out of the rest of the material. There are a 2em margin on the right and a margin of 2em on the left, but measured from the left edge of the condition. The text of the condition is indented so that its lines start exactly below the first one.

I achieve it with the help of enumitem.

documentclass{article}
usepackage{amsmath,amssymb}
usepackage{enumitem}

newlength{conditionwidth}
newenvironment{condition}[1]
 {%
  normalfont
  settowidth{conditionwidth}{normalfont#1: }%
  addtolength{conditionwidth}{2em}%
  begin{description}[
    font=normalfont,align=right,
    labelwidth=conditionwidth,
    leftmargin=conditionwidth,
    rightmargin=2em,
    labelsep=0pt,
  ]
  item[normalfont#1: ]itshapeignorespaces
 }
 {end{description}}

begin{document}

We shall often need the following fact, which we shall  usually abbreviate
into ``condition~$mathcal{P}(n_0)$'':
begin{condition}{$mathcal P(n_0)$}
  If $U=(u_n)_{nin mathbb{N}}$, $V=(v_n)_{nin mathbb{N}}$ are monotone sequences, 
  such that there exists an integer $n_0inmathbb{N}^*$ and finite subsets
  $A_{n_0},B_{n_0}$ of $E$ which verify
  [
  left{begin{aligned}
  U &subseteq A_{n_0}cup T^{n_0}(U, V )
  V &subseteq B_{n_0}cup T^{n_0}(V, U )
  end{aligned}right.
  ]
  then $U$ and $V$ are relatively weakly compact.
end{condition}
Some other text to continue the description.

end{document}

A few notes.

1. I fixed some language mistakes, in particular the use of commas;
2. I separated the two sequences into two distinct formulas, because the comma between them is text and not math;
3. intead of matrix I used aligned with a single alignment point.
4. Never use $$ in LaTeX and never leave a blank line before a math display.