So I'll keep this real short and simple. In this document on page 23 there is a list of axioms. On page 24 there is a list of theorems that come from said axioms. I can prove them all except 3 and 5. I feel like it's not hard and I just don't have enough experience with manipulating logic formulae for the answer to be clear. If someone could post the proofs for 3 and 5 I would appreciate it very much.
The list of axioms that are used:
- $\forall x \forall y\, \left[ x = y \rightarrow (\forall z: (x \in z \leftrightarrow y \in z)\,\right]$
- $\forall x \forall y\, \left[ x = y \leftrightarrow (\forall z: (z \in x \leftrightarrow z \in y) \right]$
- $\exists z \,z = \emptyset$
- $\forall x \forall y \exists z\, z = \{x,y\}$
Theorem 1.3: $\forall x \forall y \forall z \left[ (x=y \land y = z) \rightarrow x=z \right]$ (the documents says this one comes solely from the Axiom of Extensionality (axiom 2 above)
Theorem 1.5: $\forall u \forall v \forall x \forall y \left[\langle u,v \rangle = \langle x,y \rangle \leftrightarrow (u=x \land v=y)\right] $
where $\langle s,t \rangle$ is defined to be $\{\{s\}, \{s,t\}\}$.