Matrices, Equivalent Conditions

My problem is this:

State whether the following is true or false, provide a reason.

1. A**x** = O has only the trivial solution if and only if A**x** = **b** has a unique solution for every n x 1 column matrix **b.**

I am leaning towards false. The reason for this lies with a list of equivalent conditions in my book:

If A in an n x n matrix, then the following statements are equivalent:

1. A is invertible

2. A**x** = **b** has a unique solution for every n x 1 column matrix **b**

3. A**x** = O has only the trivial solution

4. A is row equivalent to

5. A can be written as a product of elementary matrices.

The reason I am leaning towards false is because the problem makes it seem that part one of it (A**x** = O** ...**) can only be true if part two (A**x = b...**) is. Perhaps I am misinterpreting however. Thanks for input.