OK, now that we see your edit with the pic we know better. That wire is a water bond for the electrical service. It is NOT being used as a ground, it is bonding the metallic water piping system in the house so that in the case of a live wire or component touching a water pipe the circuit breaker will trip. Can you see where it goes? It should go back to the main service panel.
To add, copper piping can only be used for circuit equipment grounding purposes in some very specific and strict instances. I can quote the code sections but suffice to say it is basically irrelevant since it is just as easy to run a proper circuit or ground to a panel box.
Long ago, like the 60's and 70's it was common to ground some circuits to cold water pipes, but this was found to be unsafe in many instances and has not been generally allowed for quite some time. It is not uncommon to see wires still attached to pipes in older homes.
Older 120/240V household electric dryers and cooking appliances were allowed to be "3-wire", using two hots and only a neutral, the equipment ground was allowed to be omitted and the neutral serving also as the ground. This "3-wire" allowance was removed from the code in the 90's.
Here is the applicable code section for dryers and ranges:
From the 2011 NEC
250.140 Frames of Ranges and Clothes Dryers
Frames of electric ranges, wall-mounted ovens, counter-mounted cooking
units, clothes dryers, and outlet or junction boxes that are part of
the circuit for these appliances shall be connected to the equipment
grounding conductor in the manner specified by 250.134 or 250.138.
Exception: For existing branch-circuit installations only where an equipment grounding conductor is not present in the outlet or
junction box, the frames of electric ranges, wall-mounted ovens,
counter-mounted cooking units, clothes dryers, and outlet or junction
boxes that are part of the circuit for these appliances shall be
permitted to be connected to the grounded circuit conductor if all the
following conditions are met.
(1) The supply circuit is
120/240-volt, single-phase, 3-wire; or 208Y/120-volt derived from a
3-phase, 4-wire, wye-connected system.
(2) The grounded conductor
is not smaller than 10 AWG copper or 8 AWG aluminum.
grounded conductor is insulated, or the grounded conductor is
uninsulated and part of a Type SE service-entrance cable and the
branch circuit originates at the service equipment.
contacts of receptacles furnished as part of the equipment are bonded
to the equipment.