# Subset relation

THEOREM    $$A \subseteq B \Leftrightarrow (\forall x : x \in A \Rightarrow x \in B)$$
subset_iff: (?A $$\subseteq$$ ?B) = ($$\forall$$t. t $$\in$$ ?A $$\longrightarrow$$ t $$\in$$ ?B)


Proof unavailable

THEOREM

The subset relation is reflexive.

$$A \subseteq A$$

Proof available

THEOREM

The subset relation is antisymmetric.

$$A \subseteq B \wedge B \subseteq A \Rightarrow A = B$$

Proof available

THEOREM

The subset relation is transitive.

$$A \subseteq B \wedge B \subseteq C \Rightarrow A \subseteq C$$

Proof available

THEOREM

The empty set is a subset of every set.

$$\emptyset \subseteq A$$

Proof available

THEOREM

Every set is a subset of the universal set.

$$A \subseteq \mathcal{U}$$

Proof available