Consider the system of functional differential inequalities: D(σ)[u′(t)−ℓ(u)(t)]≥0 for a. e. t∈[a,b],φ(u)≥0, where ℓ:C([a,b];Rn)→L([a,b];Rn) is a linear bounded operator, φ:C([a,b];Rn)→Rn is a linear bounded functional, σ=(σi)ni=1, where σi∈{−1,1}, and D(σ)=diag(σ1,…,σn). In the present paper, we establish conditions guaranteeing that every absolutely continuous vector-valued function u satisfying the above-mentioned inequalities admits also the inequalities u(t)≥0 for t∈[a,b] and D(σ)u′(t)≥0 for a. e. t∈[a,b].