Coset containment implies subgroup containment

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Statement

Verbal statement

If a Left coset (?) of one subgroup is contained in a left coset of another subgroup, then the subgroup is also contained in the other.

Statement with symbols

Suppose $x H \subseteq y K$ are cosets of subgroups $H, K \leq G$ . Then $H \leq K$ .

Related facts

Subgroup containment implies coset containment: If one subgroup of a group is contained in another, then every left coset of the subgroup is contained in a left coset of the other subgroup.
Nonempty intersection of cosets is coset of intersection: If the intersection of a collection of left cosets of subgroups is nonempty, it is a coset of the intersection of the corresponding subgroups.

Proof

Given: $x H \subseteq y K$ for some $x, y \in G$ .

To prove: $H \leq K$

Proof: Since $x = x e$ is in $x H$ , which is contained in $y K$ , there exists $k_{1} \in K$ such that $x = y k_{1}$ . Thus $y k_{1} H \subseteq y K$ , which implies that there exist $h \in H$ and $k_{2} \in K$ such that $y k_{1} h = y k_{2}$ . Hence $h = k_{2} k_{1}^{- 1}$ , which is in $K$ . So, $H \leq K$ .