New to proofs, need help with how to approach a beads-and-wires proof puzzle I am working on a proof puzzle about beads and wires. We are given 4 axioms about the objects you can create with beads and wires.


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*Axiom 1. You must have exactly 3 beads.

*Axiom 2. There is exactly one wire between each pair of beads.

*Axiom 3. Not all beads can be on the same wire.

*Axiom 4. Any pair of wires has at least one bead in common.


We are asked to prove the following theorem: 

Theorem. No bead can be on all wires for all possible bead-wire models. 

I am new to proofs and have only had some experience doing simple proofs based on real numbers. Therefore, I am having issues on how to rigorously use these axioms about beads and wires in a way of proving the theorem. 

So far, I can see that if one bead is on all wires, that if you follow Axiom 2 and Axiom 4 that you would need more that 4 beads if you don’t have all beads on one wire (contradicting Axiom 1). 

I am just having issues of how to rigorously represent these objects and make that into a proof instead of just a visual intuition.
Any help with how to get started would be much appreciated!

Suppose there is a bead that is on all the wires. 
By axiom $1$ there are two other beads.
By axiom $2$ there is a wire containing both these beads.
By axiom $3$ the first bead cannot be on this wire.
