Assume holds for with , and suppose, for contradiction, that there is no least such that holds. Consider .

Certainly is false, because otherwise would be our minimal element; so holds. Now, given , suppose is true for all . Then must be false for all , and so must also be false — otherwise would be our minimal element. Hence holds.

Hence by SPI, holds for all , and is false for all , contradicting the assumption that holds for some .

The converse fails: , and the well-ordering principle fails for certain ordinals. However, in any proof using SPI, one can in fact use WOP instead.