Comment by Gro-Tsen on meaning of "the singletons of $\Gamma$ are a basis for $\Gamma$"

I think this is supposed to mean that every nonempty set in class $\Gamma$ contains ($\supseteq$) a singleton in class $\Gamma$. But it could also conceivably mean that every nonempty set in class $\Gamma$ contains ($\ni$) a real in class $\Gamma$. This “basis” terminology is abominable, and totally at odds with the meaning of “basis” in other parts of mathematics; but the general idea is that $B$ being a “basis” for $P$ means $(\exists x.(P(x))) \Rightarrow (\exists x\in B.(P(x)))$.