How to prove an element does not belong to an inductive_set - isabelle

Question

Assuming I have already defined an inductive_set, for example, the inductive set "Even" such that:
inductive_set Even :: "int set"
where ZERO : "0 ∈ Even"
| PLUS :"x ∈ Even ⟹x+2 ∈ Even"
| MIN :"x ∈ Even ⟹ x-2 ∈ Even"
lemma aux : "x= ((x::int)-2) + 2" by simp
It's fairly easy to prove lemma : "2 ∈ Even" by doing a subst to replace 2 by 2-2+2
But I'm wondering how do prove lemma : "1 ∉ Even"?
Edit:
(*Javier Diaz's answer*)
lemma Even_divisible_by_2: "n ∈ Even ⟹ 2 dvd n"
by (induction rule: Even.induct) (simp, presburger+)
lemma "1 ∉ Even"
proof
assume "1 ∈ Even"
then have "2 dvd 1"
using Even_divisible_by_2 by fastforce
then show False
using odd_one by blast
qed
What would be the equivalent way to do it for 3?
lemma "3 ∉ Even"
proof
assume "3 ∈ Even"
then have "2 dvd 3"
(*how to continue?*)
qed
Thanks in advance

Answer 1

I would prove an intermediate result first, namely that each number in your inductive set is divisible by 2:
lemma Even_divisible_by_2: "n ∈ Even ⟹ 2 dvd n"
by (induction rule: Even.induct) simp_all
And then prove your result by contradiction:
lemma "1 ∉ Even"
proof
assume "1 ∈ Even"
then have "2 dvd 1"
using Even_divisible_by_2 by fastforce
then show False
using odd_one by blast
qed
I strongly recommend that you use Isabelle/Isar instead of proof scripts.
NOTE: As request by the post author, I'm adding a proof that 3 ∉ Even in the style of the above proof:
lemma "3 ∉ Even"
proof
assume "3 ∈ Even"
then have "2 dvd 3"
using Even_divisible_by_2 by fastforce
then show False
using odd_numeral by blast
qed
Alternative solution: #user9716869 provided the following more general and efficient solution based on the use of Even_divisible_by_2:
lemma n2k1_not_Even: "odd n ⟹ n ∉ Even"
using Even_divisible_by_2 by auto
lemma "1 ∉ Even" and "3 ∉ Even" and "11 ∉ Even"
by (simp_all add: n2k1_not_Even)