In Real Analysis class, a professor told us 4 claims: let x be a real number, then:
1) $0\leq x \leq 0$ implies $x = 0$
2) $0 \leq x < 0$ implies $x = 0$
3) $0 < x \leq 0$ implies $x = 0$
4) $0 < x < 0$ implies $x = 0$
Of course, claim #1 comes from the fact that the reals are totally ordered by the $\leq$ relation, and when you think about it from the perspective of the trichotomy property, it makes sense, because claim #1 says: $0 \leq x$, and, $x \leq 0$, and then, the only number which satisfies both propositions, is $x = 0$.
But I am not sure I understand the reason behind claims #2, #3 and #4. Let's analyze claim #2:
It starts saying $0 \leq x$, that means that x is a number greater than or equal to $0$, but then it says $x < 0$, therefore x is less than zero. I think no number can satisfy both propositions, as it would contradict the trichotomy property, because x is less than $0$ AND greater than or equal to $0$.
Same thing with claim #4, since $0 < x < 0$ means: $x > 0$ AND $x < 0$, and no real number satisfy both propositions at the same time. Therefore, saying $x = 0$ is the same as saying $x = 1$, or $2$, or $42$ (this is because the antecedent if always false).
Am I missing something? My professor then told us that these were axioms, but I think that axioms should not contradict well established properties (like the trichotomy property) or, at the very least, make some sense. Are claims #2 to #4 well accepted and used "axioms" in real analysis?