Four Biblical Proofs:

The First One - from the temporal order of creation,

God, creating everything from nothing, started with chaotic and inert matter and gradually proceeded toward a more perfect being. That is enough. The conclusion should be evident to a knowledgeable reader.
 

The Second One - from the quality of the material.

Clay was quite sufficient to create man, but a much more sophisticated material was needed to create woman.
 

The Third One - from behavioral traits.

Satan had to employ all of his intelligence to convince Eve, but for Adam a simple order "Eat it!" was sufficient.

 
The Fourth One - from the severity of the punishment.
 

a) The more intelligent is the culprit, the more justified is the severe punishment. Eve's punishment, subjection to Adam, is extremely humiliating for an intelligent creature.

b) This agonizing state of affairs is clearly provoked by sin. Judging from the behavior of Adam (see Third Proof above), woman was not subject to man in the Garden of Paradise right after creation.

Two Philosophical Proofs:

The Fifth One - from external appearance:

Man occupies an intermediate position between animals and angels. Who look more like angels, men or women! Any further comment is superfluous.
 

The Sixth One - from intuition.

St. Thomas Aquinas admitted that women have a better intuitive knowledge than men, therefore, they are closer to angels in the dynamism of their minds,

One Theological Proof:

The Seventh One - from the humility of Our Lord.

In order to give us an example of humility, Our Lord took on the body of man.
 
 

It is possible to construct a further proof - a mariological one. But do I really have to discover everything. Let others use their brains, too!