How do I prove by contradiction that Art's taller than Bob?

Question

From: Philip Johnson-Laird BA PhD Psychology (UCL), Stuart Professor of Psychology Emeritus at Princeton. (Author isn't a logician.) How We Reason (1st edn 2008). p. 274.

Experiments have therefore posed relational problems of this sort:

[1.] Art is taller than Cal to a greater extent than Bob is taller than Dan.
[2.] Bob is taller than Dan to a greater extent than Cal is taller than Dan.

What is the order of the four individuals in terms of their height? The answer in this case, with the tallest first, is: Art, Bob, Cal, Dan. One strategy for solving the problem is to realize that the second premise establishes the order:

[3.] Bob Cal Dan

and then to use the first premise to add Art in his appropriate place. If you doubt whether Art is taller than Bob, you should work out the consequences of the supposition that they’re the same height. It is then impossible for the first premise to be true, because
[4.] Bob would also have to be taller than Cal to a greater extent than Bob is taller than Dan, and that would conflict with the order of the three of them above.

I understand that the author is proving by contradiction that A's height > B's.

5. Doesn't 3 imply, and thus, substantiate 4?

6. If yes to 5, how does 4 beget the contradiction?

Answer 1

The first statement translated into an inequality is $$ A-C > B-D > 0. \tag{1}$$ The second statement translates into an inequality $$ B-D > C-D > 0. \tag{2}$$ The first inequality implies, using algebra, that $$ A-B > C-D. \tag{3} $$ The second inequality implies, using algebra, that $$ B-C > 0. \tag{4} $$ Combining $(2)$ and $(3)$ together gives $$ A-B > C-D > 0. \tag{5} $$ Finally, $(4)$ and $(5)$ together implies $$ A > B > C > D. \tag{6} $$ There is no need for proof by contradiction here.