Why are homotopy spheres spin-cobordant in dimensions divisible by 4?

Question

Manifolds are assumed to be smooth having dimensions $\geq5$. As usual, $\Omega^{Spin}_{*}$ denotes the spin bordism ring and $\Omega^{SO}_{*}$ the oriented bordism ring.

Let $\Sigma^n$ be a closed homotopy sphere. If $n\equiv0(4)$ then $$[\Sigma]=[\emptyset]\in\Omega^{Spin}_{n},$$ what can be followed from the following two facts:

• The canonical ring homomorpishm of $\Omega_*^{Spin}\rightarrow\Omega^{SO}_{*}$ is injective in dimensions $n\equiv0(4)$.
• It holds $[\Sigma]=[\emptyset]\in\Omega^{SO}_{n}.$

I asked myself

1. Can one derive the assertion more easily with another strategy?
2. If not, can one proof the two facts without using heavy machinery? How?
3. If questions 1. and 2. were answered negative, can you give me references, from which the assertion follows?