Maschler's bargaining set-an incomplete step in a proof

I have a problem with the concept of the bargaining set which is given below in some detail. Let $$N=\{1,\ldots,n\}$$ be a set of players and $$v:2^N\to \mathbb{R}$$ a superadditive game (meaning $$S,T \subseteq N ,\,S \cap T=\emptyset \Longrightarrow v(S)+v(T)\le v(S \cup T).$$) in coalitional function form. For $$y\in\mathbb{R}^C$$ and $$S\subseteq N$$ write $$y(S)$$ for $$\sum_{i\in C}y_i.$$ So $$y(\emptyset)=0.$$ Write $${\cal S_{kl}}=\{T \subseteq N\,|\,k \in T,\, l \notin T\}.$$ Let $$x\in \mathbb{R}^D$$ be an individually rational imputation with $$x(D)=v(D)$$. What is the relationship between the following two formulas-I'd like them to be equivalent $$\forall y\in \mathbb{R}^C$$:

$$(\exists D \in {\cal S_{lk}})\big(D \cap C=\emptyset \;\wedge\; x(D)\leq v(D)\big) \quad\vee$$ $$\Big((\forall i \in C)(y_i > x_i)\quad\wedge$$ $$(\forall D \in {\cal S_{lk}})\big(x(D \setminus C)+y(D \cap C) > v(D)\big)$$ $$\ \Longrightarrow \ y(C)>v(C)\Big).$$

and

$$(\exists D \in {\cal S_{lk}})\big(D \cap C=\emptyset \;\wedge\; x(D)\leq v(D)\big) \quad\vee$$ $$\Big((\forall i \in C)(y_i \geq x_i)\quad\wedge$$ $$(\forall D \in {\cal S_{lk}})\big(x(D \setminus C)+y(D \cap C) \geq v(D)\big)$$ $$\ \Longrightarrow \ y(C)\geq v(C)\Big).$$

We may assume that the first part before $$\vee$$ which is the same in both is not satisfied. The only difference is by replacing some $$>$$ with $$\geq.$$ The problem arises when $$D\cap C=\emptyset,$$ otherwise they are equivalent by increasing coordinates of $$y$$ and slight considertaions. Perhaps here may be used the superadditivity of $$v$$?

I believe that the second is equivalent to this

$$\Big(k,l\in N \;\wedge\; k \neq l \;\wedge\; C \in {\cal S_{kl}} \quad \Longrightarrow$$

$$(\exists D \in {\cal S_{lk}})\big(D \cap C=\emptyset \;\wedge\; x(D)\leq v(D)\big)\quad\vee$$

$$\min\big\{y(C)\,|\, x(D \setminus C) +y(D \cap C) \geq v(D),\, D \in {\cal S_{lk}},\, y_i \geq x_i,\, i \in C \big\} \geq v(C)\Big),$$ which is my aim to obtain from the first displayed formula with strict inequalities which are problematic to obtain these weak inequalities and $$\min$$. The entire thing here is to obtain weak linear inequalities connected by Boolean connectives that determine the Maschler bargaining set which is a quite interesting concept in cooperative game theory. The formulas are supposed to describe the bargaining set $$\cal M$$, which is such that there is no justified objection of a player $$k$$ to $$l$$. And it is of the form finite union of polytopes in $$\mathbb{R}^N.$$