The earth, moon + sun and the rest of the solar system were created from the same cloud of gas and dust.
The hydrogen in the sun was created in the big bang, although some of it would have also been in the outer unburnt envelope of early generations of stars.
Everything in the solar system larger than hydrogen was created in those earlier generations of stars and all the elements on Earth heavier than iron were created in supernovae.
So in a way, yes we are formed from the remnants of a supernova ( or actually many many sn ) but the Earth and moon weren't individually shot-out of a supernova.