I prove it with logic, not observable facts. Logic is the foundation of truth and knowledge, and we could not make any sense of the world (or even coherently construct sentences) without logic. It is necessary to our understanding of the world. A simple proof by contradiction shows that "all truths are subjective" contradicts itself (because, if it were true, it would be an objective truth and therefore make itself false). If an argument is contradictory, basic logic tells us to slap a "not" in front of it, so: "not all truths are subjective"
must be true. This sentence is logically equal to: "there exists at least some objective truth." See, logic.
With that said, I'll go further and claim that all truths are objective. For something to be true, it must be objective. The difference between objective and subjective is often misunderstood, but pretty simple really. Subjective beliefs are ones you cannot object or dispute because they're
unmeasurable. Measuring can be as simple as a 0 or 1 value: false and true, or it can be a matter of physics (mass, displacement, temperature, or time), or some other mathematical measurement. Objective truths, then, are ones that can be measured.
Truths are by definition objective, unless you use the phrase "subjectively true," or some variation thereof, but then it becomes a different subject. It's just written into the definition of truth that it must be objective: in the 0 or 1 sense of things.