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The ⊢ symbol has not changed; it means that the formula to which it applies is asserted to be true. ⊃ is logical implication, and ≡ is logical equivalence. Λ is the empty set, which we write nowadays as ∅. ∩ ∪ and ∈ have their modern meanings: ∩ and ∪ are the set intersection and the union operators, and x∈y means that x is an element of set y.
The remaining points are semantic. α and β are sets. 1 denotes the set of all sets that have exactly one element. That is, it's the set { c : there exists a such that c = { a } }. Theorems about 1 include, for example:
that Λ∉1 (∗52.21),
that if α∈1 then there is some x such that α = {x} (∗52.1), and
that {x}∈1 (∗52.22).
2 is similarly the set of all sets that have exactly two elements. An important theorem about 2 is ∗54.3, which says
∗54.3.⊢2=α^{(∃x).x∈α.α−ι‘x∈1}.