I do not understand your question: a set is infinite if and only if there is a bijection between it and one of its strict subsets. As formal as it gets. Edit:

I do not understand your question: a set is infinite if and only if there is a bijection between it and one of its strict subsets. As formal as it gets. Edit: (not intending to be harsh, just that I probably lack some context to properly understand what you are asking).