During the first few pages of Spivak's Calculus (Third edition) in chapter 1 it mentions six properties about numbers.
(P1) If $a,b,c$ are any numbers, then $a+(b+c)=(a+b)+c$
(P2) If $a$ is any number then $a+0=0+a=a$
(P3) For every number $a$, there is a number $-a$ such that $a+(-a)=(-a)+a=0$
(P4) If $a$ and $b$ are any numbers, then $a+b=b+a$
(P5) If $a,b$ and $c$ are any numbers, then $a\cdot(b\cdot c)=(a\cdot b)\cdot c$
(P6) If $a$ is any number, then $a\cdot 1=1\cdot a=a$
Then it further states that $1\neq 0$. In the book it says that it was an important fact to list because there is no way that it could be proven on the basis of the $6$ properties listed above - these properties would all hold if there were only one number, namely $0$.
1) How does one rigorously prove that $1\neq0$ cannot be proven from the $6$ properties listed?
2) It says that "these properties would all hold if there were only one number, namely $0$." Is a reason as to why this is explicitly mentioned is to avoid this trivial case where we only have the number $0$? Is there another deeper reason as to why this sentence was mentioned in relation to $1\neq 0$?
NB: Can someone please check if the tags are appropriate and edit if necessary? Thanks.