a shred of proof

The slightest bit of evidence or proof. Usually used in negative constructions. You can't just come in here without a shred of proof and start throwing around accusations like that! Your honor, it is clear that the prosecution doesn't have a shred of proof to back up their claims. That guy can say whatever he wants—without a shred of proof, no one will believe him.

without a shred of proof

Without having even a modicum of evidence or proof. You can't just come in here without a shred of proof and start throwing around accusations like that! Your honor, it is clear that the prosecution has conjured these charges against my client from thin air, without a shred of proof to back up their claims.