MHB Prove No Bijection between x and x^+

[Andrei1]

Let $$x$$ be a natural number (set). How to prove that there is no bijection between $$x$$ and $$x^+$$, where $$x^+=x\cup\{x\}$$? Then I can show that $$\mathrm{card}\,x<\mathrm{card}\,x^+.$$ I know that $$x\notin x.$$

 

[Andrei1]

$$\mathrm{card}\,x<\mathrm{card}\,x^+.$$

I took this problem from Zorich "Mathematical analysis". Zorich spoke just twice about infinity before this problem: 1. Dedekind's definition, 2. Axiom of infinity. Suppose there is a bijection from $$x$$ to $$x^+.$$ Since $$x\subset x^+$$, then $$x$$ is infinite by 1, which is false, but I don't know how to prove it.