Strong paraconsistency, also called dialetheism, demands a thorough revision of the classical ideas of negation, opposition and contradiction, by claiming that some contradictions hold, are true, and it is rational to accept and assert them. Although such a position is naturally portrayed as a challenge to the Law of Non-Contradiction (LNC), all the main formulations of the Law are not disputed by a dialetheist, in the sense that she is committed to accept them by her own theory. Her dialetheic attitude is expressed by typically accepting, and asserting, both the usual versions of the Law, and sentences inconsistent with them. The aim of this paper is to develop a formulation of the Law which appears to be unquestionable, in the sense that strong paraconsistentists are committed to accept it without also accepting something inconsistent with it, on pain of trivialism - that is to say, on pain of lapsing into the position according to which everything is the case. This will be achieved by characterizing a negation operator via the primitive intuition of material opposition, or content exclusion, which I claim to be shared by paraconsistent logicians and dialetheists, too. Strong paraconsistentists ask us to stop using 'not' (as well as 'true') as an exclusion-expressing device, because 'not- ' is insufficient by itself to rule out (and ' is true' is insufficient by itself to rule out that is also false). However, the dialetheic account of the pragmatic notions of acceptance and rejection shows that strong paraconsistentists do believe in the impossibility of some couples of 'facts', or 'states of affairs', simultaneously obtaining; or, equivalently, that they assume that some properties or states of affairs, such as x's accepting and x's rejecting the same sentence, are materially opposed to each other. By means of an exclusion-expressing negation characterized via the intuitive notion of material opposition, we may establish a minimal formulation of the LNC, in the sense of a version on which both the orthodox friend and the paraconsistent foe of consistency can agree. All of this shall not constitute a cheap victory on dialetheists: we may just learn that different things have been historically conflated under the label of 'Law of Non-Contradiction'; that dialetheists rightly attack some formulations of the Law, and orthodox logicians and philosophers have been mistaken in assimilating them to the indisputable one.