Typical - and spectacular - missing the point though.
First, he's tweeting in support of TRAs - so the context is about women losing "rights" relative to (in his view) an even more under-privileged minority. The GCs may disagree with that premise, but this is the context in which he's made that statement and context is important. Taking the tweet at face value and applying it more broadly to a range of other issues is highly selective misquoting and demeans anyone trying to make a serious point.
Second, of course women's rights were won at the expense of men's rights. The fact that this previous allocation of rights was unfair, unjust and ultimately not in the long-term interests of society and the economy doesn't change that. A PP asked "did men lose the right to vote - No?" but missed the fairly obvious point that after women's suffrage was achieved, men's votes only accounted for 50% of the theoretical maximum votes cast. Oh ... and men also, eventually, started losing their "right" to single sex spaces (e.g. Royal & Ancient Golf Club, St Andrews etc.).
Why do people who want change feel the need to pretend it comes at no cost. There's always a cost. The winning argument ought to be, "those costs are worth it because the benefits are even greater" - not "there's no costs" or even worse "because the costs are borne by people I don't like, it's OK to disregard them".
I wonder if the reason TRAs get under the skin of so many feminists is because they really do expose some of the fault lines in feminism? Sex didn't matter when it was about rising up in the workplace and getting seats at the top table at board meetings (even though, patently and demonstrably, sex does and always has mattered hugely in most professions). But now that the issue of the day is about access to toilets suddenly sex does matter again? Where's the logic? But sure, if society wants to change its mind (or evolve) then the corollary is that people ought to freely admit that there really was no justification at all for all those court cases and furore about members clubs in London - and that men losing "rights" was a necessary and worthwhile price to pay so that women could gain equality? Isn't that what Billy Bragg's tweet actually says?