Proof of Levenshtein distance

In the article Levenshtein distance Wikipedia says about the proof of invariant that:

This proof fails to validate that the number placed in d[i,j] is in fact minimal; this is more difficult to show, and involves an argument by contradiction in which we assume d[i,j] is smaller than the minimum of the three, and use this to show one of the three is not minimal.

So how can I show that d[i,j] is actually minimal? (informal proof is ok)

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