$\forall x\forall y(x<y \to y\not<x)\iff \forall x\forall y((x\le y \text{ and } x\neq y)\to (y\not\leq x\text{ or }x=y))$

Question

This is how I got this equivalence:

$$x < y → y≮x ↔ $$

$$x < y → ~(y < x) ↔ $$

$$(x≤y ∧ x≠y) → ~(y≤x ∧ x≠y) ↔$$

$$(x≤y ∧ x≠y) → (~y≤x ∨ ~x≠y) ↔$$

$$(x≤y ∧ x≠y) → (y≰x ∨ x=y)$$

I would like to know if this is correct