Can we have an antivirus that detects all viruses?
No math needed. Experiment time. Lets arrange a pool of viruses, a pool of systems-that-try-to-defend-themselves and mechanisms that constantly change/improve both parties.
Give them a planet. Wait billions of years.
Do we still have viruses?
For me the sheer number of possibilities already tested is far beyond satisfactory.
About "proving"
Here I'd like to explain why I've drastically reframed your question.
The direct answer to your question "Has it been mathematically proven that <statement about real world>" is: No, mathematicians have never proven your statement about real world (or any other such statement). By real world I mean the one that you see when you look out your window.
You can only prove anything in axiomatic systems ("worlds" that are ruled by a set of axioms - where axioms themselves are defined as unprovable). I don't know axioms that rule the real computer viruses, so I cannot prove anything about these.
I'll try to predict reality and I'll be often correct, sure. But I will use falsifiable theories for that, not proofs.