To answer this question properly, the notions used in the question should first be properly defined, since in the literature one name is often associated with more than one notion.
Be it as it may, if by 'binding energy of electron' is meant ionization energy, then the answer to the qestion is in general in the negative. To appreciate this, the size of the quasi-particle energy gap EG in an N-particle system is equal to
(1) EG = I - A,
where
(2) I = EN+1;0 - EN;0
is the first ionization energy, and
(3) A = EN;0 - EN-1;0
the electron affinity. Above EM;0 denotes the ground-state energy of the M-particle system, where M = N-1, N, N+1. In order for EG = I, that is A = 0, one must have EN;0 = EN-1;0,which in general is not the case. For a conventional semiconductor or insulator, I is the lowest conduction-band energy and A the highest valence-band energy.
Lastly, the binding energy of a condensed-matter system is often defined in terms of the difference in the ground state energies of the system in the condensed phase and in the dilute gaseous phase. For instance, the binding energy per unit cell is the difference in the ground-state energy per unit cell of the system under discussion in condensed phase and the sum of the ground-state total energies of the atoms that occupy the relevant unit cell (with each atom considered in isolation). The relationship between this binding energy and the quasi-particle energy gap is a non-trivial one, since here one is dealing with systems with very different single-particle Hilbert spaces. To appreciate this, just think of the fact that while the binding energy for a metallic solid is certainly non-vanishing, the energy gap in this solid is vanishing (at least, it is microscopically small).
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