I need assistance trying to simplify a logic statement

Question

The statement I am trying to simplify is: $\lnot(p\lor\lnot q)\to(p\to(p\land\lnot p))$

First thing I did was use the Material Implication Law resulting:
$\lnot\lnot(p\lor\lnot q)\lor(\lnot p\lor(p\land\lnot p))$
$(p\lor\lnot q)\lor(\lnot p\lor(p\land\lnot p))$

Then I distributed the right side resulting in:
$(p\lor\lnot q)\lor((\lnot p\lor p)\land(\lnot p\lor\lnot p))$

And since $(\lnot p\lor\lnot p)=\lnot p$:
$(p\lor\lnot q)\lor((\lnot p\lor p)\land\lnot p)$

And since $\lnot p\lor p=\top$:
$(p\lor\lnot q)\lor(\top\land\lnot p)$

Up to there is where I feel stucked and I am not sure if I did it correctly and would like appreciate some feedback if its correct or wrong and what can I do to solve it better. Thanks in advance!

Answer 1

Given expression $$ \neg (p \vee \neg q) \rightarrow ( p \rightarrow (p \wedge \neg p))$$ $$ = \neg (p \vee \neg q) \rightarrow ( \neg p \vee F) $$ $$ = ( p \vee \neg q) \vee \neg p = (T \vee \neg q) = T$$ As you are stuck after this: $$( p \vee \neg q) \vee ( T \wedge \neg p)$$ On using $ T \wedge \neg p = \neg p$, the statement becomes, $$ p \vee \neg q \vee \neg p$$ On using $\neg p \vee p = T$ , the statement becomes, $$ T \vee \neg q = T$$