I already asked this question here, but unfortunately no acceptable answers were provided. Also, I hope this is not too basic for MathOverflow.

I'm studying on K. Hrbacek and T. Jech, Introduction to Set Theory. In the third chapter, they prove the usual properties of the strict ordering on natural numbers in the following way:

- They prove transitivity using induction.
- They prove irreflexivity using induction and transitivity.
- They prove asymmetry by contradiction and using transitivity and irreflexivity.

There is nothing wrong with this, but I'd like, if possible, to prove the previous properties (in particular irreflexivity and asymmetry) independently and using induction only.

For example, let $\phi(x)$ be $x\not\in x$. Of course $\phi(0)$ is true ($0\not\in 0$ since $0$ is the empty set). Now I should suppose that $\phi(x)$ is true and show that $\phi(x+1)$ is true, too. $\phi(x+1)$ stands for $x+1\not\in x+1$, that is $x+1\not\in x$ and $x+1\not=x$. But here I cannot go on, mainly because $x+1$ is at the left of the relations symbols.

$x+1$ is defined as $x\cup\{x\}$.

Any hint? Thank you.