How to prove that all maximal independent sets of a matroid have the same cardinality.
Provided a matroid is a 2-tuple (M,J ) where M is a finite set and J is a family of some of the subsets of M satisfying the following properties:
If A is subset of B and B belongs to J , then A belongs to J , If A, B belongs to J , |A| <= |B|, and x belongs to A - B, then there exists y belongs to B - A such that (B U {x})- {y} belongs to J. The members of J are called independent sets.
Assume by way of contradiction that you have two maximal independent sets $I_1$ and $I_2$ such that $|I_1| > |I_2|$. Can you apply the third axiom you listed to get a contradiction?