My interpretation of Von Neumann–Morgenstern utility theorem

Assume:

We have systems of distribution.

In S1 your probability of getting A is Ps1a and your probability of getting B is Ps1b.

and Ps1a+Ps1b=1

In S2 your probability of getting A is Ps2a and your probability of getting B is Ps2b.

and Ps2a+Ps2b=1

1) You either prefer S1 to S2 or S2 to S1 or you are indifferent

2) If for you S1>S2 and S2>S3 then S1>S3

3) You preference between S1 and S2 and S3 is continious

4) If you prefer S1 to S2 (S1>S2)

and you have a chance of p to get involved in S1 when you must inevitably engage in S3 with a chance of (1-p)

You would prefer this scenario compared with getting involved in S2 when you must inevitably engage in S3 with a chance of (1-p)

p.S1+(1-p).S3 > p.S2+(1-p).S3

They prove that

there exists a utility function U such that if you prefer S1 to S2

S1>S2

U(S1) >U(S2)

Your utility U under S1 will be U(S1)=U(Ps1a.A+Ps2b.B)

There is also U under which U(S2)=U(Ps2a.A+Ps2b.B)

This means that Ps1a.U(A)+Ps2b.U(B)>Ps2a.U(A)+Ps2b.U(B)

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