Quantification of Entropy and the 2nd Law of Thermodynamics

[transmini]

All of my information comes from my current chemistry class, I just want to know where I either may have misread, misinterpreted, or was mistold information.

With the 2nd Law of Thermodynamics we can say $$[1]\space\space \Delta{S}_{universe} = \Delta{S}_{surroundings} + \Delta{S}_{system} > 0$$
Then using $$[2]\space\space \Delta{S}_{system} = \frac{q_{rev}}{T}$$
with (at constant pressure) $$[3]\space\space q = \Delta{H}$$
and (from the 1st Law of Thermodynamics) $$[4]\space\space \Delta{H}_{system} + \Delta{H}_{surroundings} = 0$$
we can get $$[4]\space\space \Delta{S}_{system} = \frac{\Delta{H}_{system}}{T}$$ $$[5]\space\space \Delta{S}_{surroundings} = \frac{\Delta{H}_{surroundings}}{T} = \frac{-\Delta{H}_{system}}{T}$$.
Substituting these back into [1] would give $$[6] \space\space \Delta{S}_{universe} = \frac{-\Delta{H}_{system}}{T} + \frac{\Delta{H}_{system}}{T} > 0$$
which simplifies to $$0 > 0 $$
which is not true.

I'm assuming I did something wrong with this, but cannot figure out what. Where is the issue?

 

[Ygggdrasil]

The second law states that $\Delta S_{universe} \geq 0$. Specifically, for reversible processes (you are assuming reversibility because you've made q_rev = q), ΔS_universe = 0.