Elementary set shift ($S_{ij}$)

For $i<j$, replace a member $A$ containing $j$ but not $i$ by $(A\setminus \{j\})\cup \{i\}$ only when the replacement is absent. An elementary set shift preserves family size and the intersecting family property. The inclusion $\partial (S_{ij}\mathcal{F})\subseteq S_{ij}(\partial \mathcal{F})$ shows that it cannot increase the lower shadow.

A set family is shifted if every elementary set shift leaves it unchanged. For a uniform set family, this is equivalent to downward closure under componentwise comparison of the increasing lists of its elements. Repeated elementary set shifts terminate at a shifted set family.

For a member of a shifted set family that is an intersecting family, find the first prefix with more selected than unselected positions. Reflect membership in that prefix. The result has one fewer element and lies in the lower shadow; the first prefix with more unselected positions gives the inverse. This injection proves the intersecting shadow lemma.