Let F be a binary clutter. We prove that if F is non-ideal, then either F or its blocker b(F) has one of L_7, O_5, LC_7 as a minor. L_7 is the non-ideal clutter of the lines of the Fano plane, O_5 is the non-ideal clutter of odd circuits of the complete graph K_5, and the two-point Fano LC_7 is the ideal clutter whose sets are the lines, and their complements, of the Fano plane that contain exactly one of two fixed points. In fact, we prove the following stronger statement: if F is a minimally non-ideal binary clutter different from L_7,O_5, b(O_5), then through every element, either F or b(F) has a two-point Fano minor.
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