The rules don't explicitly state
FWIW the rules for constructing a wonder explicit mention that you do not pay the cost of the Age card:
the rules wrote:
3. Construct a Wonder
You pay the cost of the Wonder (not the one of the Age card), then place your Age card face down partially covered by the Wonder card which is being constructed. The used card has no effect, it is simply used to show that the Wonder has been built.