Statement is true but contrapositive is false?

Question

It seems that the contrapositive of a true statement is true. Consider the following:

1. $A,B,C,K\in \mathbb{N}$
2. $\exists A,B,C: A^K+B^K=C^K→K≤2$
3. Take the contrapositive: $$K>2→∀A,B,C:A^K+B^K<C^K∨A^K+B^K>C^K$$

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However, can the contrapositive of a true statement be false? I ask because of the following:

1. $A,B,C,D,K\in \mathbb{N}$
2. $\exists A,B,C,D: A^K+B^K+C^K=D^K→K≤2$
3. Take the contrapositive: $$K>2→∀A,B,C,D:A^K+B^K+C^K<D^K∨A^K+B^K+C^K>D^K$$ But this seems false due to the existence of Plato’s Number.

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Another example where the contrapositive of a true statement seems false is the following:

1. $A,B,C,K\in \mathbb{N}$
2. $\exists A,B,C: A^K+B^K=C^K→K≤1$
3. Take the contrapositive: $$K>1→A^K+B^K<C^K∨ A^K+B^K>C^K$$ But this seems false too.

Answer 1

Too long to post as a comment, but note that in the below, (1) and (1C) are logically equivalent, as are (2) and (2C).

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$∃A,B,C,D: A^K+B^K+C^K=D^K→K≤2\tag1$

$$∃A,B,C,D\:( A^K+B^K+C^K=D^K)→K≤2\tag1$$ and $$∃A,B,C,D\:( A^K+B^K+C^K=D^K→K≤2)$$ are logically inequivalent, so please replace that confusing colon with parentheses. Fixing this and making explicit your implicit universal quantification of $K:$ $$\forall A,B,C,D,K\:( A^K+B^K+C^K=D^K→K≤2).\tag1$$

Contrapositive: $$\forall A,B,C,D,K\:( K>2\to A^K+B^K+C^K\ne D^K).\tag{1C}$$

$∃A,B,C: A^K+B^K=C^K→K≤1\tag2$

$$\forall A,B,C,K\:(A^K+B^K=C^K→K≤1)\tag2$$

Contrapositive: $$\forall A,B,C,K\:(K>1→A^K+B^K\ne C^K).\tag{2C}$$