Propositional logic : $¬A \land (A \lor B) \vdash B$

Question

First time posting here. I'm going through some examples and stumbled upon $¬A \land (A \lor B) \vdash B$.

I'm hoping someone can help me with understanding it. This is what I've done so far, mostly I don't know how to continue from line 9 onwards taking into consideration the first part is correct.

TIA

$¬A \land (A \lor B) \vdash B$

1.  ¬A ^ (A v B)        hyp
2.      |¬B     ass
3.      |¬A     1, ^E1
4.  ¬B => ¬A        2-3, =>I              
5.      |¬B     ass
6.      |   |¬A sub ass
7.      |   |¬B copy 5.
8.      |¬A => ¬B   6-7, =>I
9.      |   |¬A sub ass
.
.
.
.
.
.
a.      |¬A => B
b.      |¬¬A        8,a, ¬I                     
c.      |A      b, ¬E
d.  ¬B => A         5-c, =>I
e.  B           4,d ¬I          

Update. Tried it like this. Does it make sense? The only issue I have is with line 8 ...

1.  ¬A ^ (A v B)        hyp
2.      |¬B             ass
3.      |A v B          1, ^E2
4.  ¬B => A v B         2-3, =>I              
5.      |¬B             ass
6.      |¬A             1, ^E1
7.      |¬A v ¬B        6, vI1
8.      |¬(A v B)       (Taking not common)
9.  ¬B => ¬(A v B)      5-8, =>I
10. ¬¬B                 4,9 ¬I
11. B                   10, ¬E

Answer 1

First step: we have to "unpack" the premise : $\lnot A \land (A \lor B)$ using $\land$-elim.

Thus, the derivation must start with :

1) $\lnot A \land (A \lor B)$ --- premise

2) $\lnot A$ --- from 1) by $\land$-elim

3) $(A \lor B)$ --- from 1) by $\land$-elim.

Then we have to apply $\lor$-elim to 3) :

4) $A$ --- assumed from 3) for $\lor$-elim

5) $B$ --- form the contradiction of 2) with 4) by Ex falso

6) $B$ --- assumed from 3) for $\lor$-elim

Having derived $B$ under both assumptions 4) and 6), we have

7) $B$ --- from 3) and 4)-5) and 6) by $\lor$-elim, discharging temporary assumptions 4) and 6).