Can Logical Inference Rules Prove This Propositional Argument?

[MarcL]

Homework Statement

For each of the premise-conclusion pairs below, give a valid step-by-step argument ( proof ) along with the name of the inference rule used in each step

premise { ¬ p → r ∧ ¬ s , t → s , u → ¬p , ¬w , u ∨ w } conclusion : ¬t ∨ w

Homework Equations

All the inference rules, Modus ponens, Modus tollens, etc...

The Attempt at a Solution

[/B]I tried by using the w term but it didn't work so I did this:

1) u → ¬ p Prmise
2) ¬p → r ∧ ¬s ass
3) u → r ∧ ¬ s

However I seem stuck

 

[Svein]

1. u⇒¬ p
2. ¬ p⇒r ∧ ¬ s
3. ∴u ⇒ r ∧ ¬ s
4. t⇒s
5. ∴¬ s⇒¬ t
6. u ∨ w
7. ¬w
8. ∴u
9. ∴r ∧ ¬ s
10. ∴¬ s
11. ∴¬ t
12. ¬w
13. ∴¬t ∨ w

 

[MarcL]

you're allowed to re-use ¬w>?

 

[Svein]

you're allowed to re-use ¬w>?

Why not? A premise is a premise.