The authors, Jeff Landauer and Joseph Rowlands, claim that "Existence Exists", that is, "there is something, as opposed to nothing", is an example of something "we cannot disprove by any other statement". They call this an axiom and note the following about axioms:
A true axiom can not be refuted because the act of trying to refute it requires that very axiom as a premise. An attempt to contradict an axiom can only end in a contradiction.
Here is their proof for Existence Exists being such an axiom:
At the core of every thought is the observation that "I am aware of something". The very fact that one is aware of something is the proof that something in some form exists -- that existence exists -- existence being all that which exists. Also, to grasp the thought, "I am aware of something," you must be conscious. Existence is axiomatic because it is necessary for all knowledge and it cannot be denied without conceding its truth. To deny existence is to say that something doesn't exist. A denial of something is only possible if existence exists.
To put this in other words, attempt to disprove this by contradiction: assume nothing. The very act of assuming is something contradicting nothing. For nothing to be true, one cannot even assume nothing. That is, one cannot even begin to disprove the axiom that Existence Exists. So if one struggles in any way to disprove this axiom, that struggle shows the axiom is true.
One is welcome to doubt this, but the very act of doubting is evidence that "there is something as opposed to nothing".
Landauer, J., Rowlands, J. Importance of Philosophy. Retrieved on September 19, 2019 at http://www.importanceofphilosophy.com/