How does one prove the following using natural deduction: $a ∧ ¬a \vdash b ∧ ¬b$

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I am trying to prove the following using natural deduction and this is what I have so far. I am not sure, however, if this is entirely correct.

If I could get some verification and be pointed to the right direction, that would be great.

    a ∧ ¬a |- b ∧ ¬b
1     a ∧ ¬a     assump 0
2     b |- a ∧ ¬a 
2.1   b          assump 2
2.2  ¬b         ¬I 2.1
3     b ∧ ¬b    ∧I 2.1, 2.2
4    ¬b |- a ∧ ¬a 
4.1  ¬b          assump 4
4.2   b         ¬E 4.1
5     b ∧ ¬b    ∧I 4.1, 4.2

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