If those who claim god exisdts won't define their god, so as to leave
nothing to aim for when debunking any claim you put forth,
yes.

Which god do I debunk? The Greek Philosopher perfect god from who's
perfection we derive omnipotence, omnibenevolence and other omni-attributes?
Which all soon contradict each other?

Or the crude, vicious, genocidal and stupid god of the old testament tall
tales?

Or both?

If you wish you avoid a debunked god by refusing to define your god, it
only delays the debunking god so deserves.

A[a, b] A[c, d] ( [a, b] < [c, d] <-> Ex(b pp x /\ x pp c))

[[a, b]] < [[c, d]] :<-> Ex(b pp x /\ x pp c))

In addition, we need an inclusion relation. With respect to the natural
ordering on Ord, we might write

A[a, b] A[c, d] ( [a, b] nat-include [c, d] <-> (Ax(x pp a -> x pp c) /\
Ax(x pp d -> x pp b)))

Let me reformulate your _definition_ this way:

[[c, d]] c [[a, b]] :<-> Ax(x pp a -> x pp c) /\
Ax(x pp d -> x pp b)

ok?

so that monotonicity

(([a, b] nat-include [c, d]) /\ ([a, b] < [x, y])) -> [c, d] < [x, y]

[[c, d]] c [[a, b]] & [[a, b]] < [[x, y]] -> [[c, d]] < [[x, y]]

((([x, y] < [a, b]) /\ ([a, b] nat-include [c, d]) -> [x, y] < [c, d]

[[x, y]] < [[a, b]] & [[c, d]] c [[a, b]] -> [[x, y]] < [[c, d]]

and convexity

( ([a, b] < [c, d]) /\ ([c, d] < [e, f]) /\ ([x, y] nat-include [a, b])
/\ ([x, y] nat-include [e, f]) ) -> ([x, y] nat-include [c, d])

are preserved.

[[a, b]] < [[c, d]] & [[c, d]] < [[e, f]] &
[[a, b]] c [[x, y]] & [[e, f]] c [[x, y]] -> [[c, d]] c [[x, y]]

However, taking lead from Cohen forcing and the construction of Suslin
trees from Suslin lines, we invert the sense of the natural inclusion
for intervals. That is,

A[a, b] A[c, d] ( [a, b] include [c, d] <-> [c, d] nat-include [a, b])