I noticed your core definition is wrong. Here’s the right one:
A matching μ is in the core if there exists no matching ν and a coalition T ⊆ m ∪ w such that v(i)Piμ(i) and v(i) ∈ T for any i ∈ T.
If that definitions holds, then the set of stable matchings is equal to the core.
First, suppose μ is individually irrational. This implies that μ is dominated by a coalition, and if its unstable due to a woman and a man such that m>wμ(w) and w>mμ(m), then it is dominated by the coalition (m,w) by any matching μ’ with μ’(m)=w.
On the other hand, suppose μ isn’t in the core. Then, μ is dominated by another matching μ’ due to a coalition. Now, if μ isn’t individually rational, μ’(w) ∈M for all w in A. This is by definition, since every woman prefers μ’(w) to μ(w).
Then, if w ∈A and m= μ’(w), this means that m prefers w to μ(w) and μ is blocked by (m,w).
Intuitively, if the matching μ isn’t stable, then it’s not in the core. This is, another match μ’ exists such that dominates μ.