This proof makes no sense to me.
The theorem to be proved is
Theorem 44. {x,y} = {u,v} → (x = u & y = v) V (x = v & y = u)
where {x,y} and {u,v} are sets with exactly two members, which can be either sets or individuals. The proof relies on:
Theorem 43. z [itex]\in[/itex] {x,y} z = x V z = y.
The given proof is:
"By virtue of Theorem 43
and thus by the hypothesis of the theorem
Hence, by virtue of Th. 43 again
(1)
By exactly similar arguments
(2)
(3)
(4)
We may now consider two cases.
Case 1: x = y. Then by virtue of (1), x = u, and by virtue of (2), y = v."
This is where I got lost. Couldn't I just as easily argue that x = v by virtue of (2) and y = u by virtue of (1)? Or by virtue of (3) and (4)? What's the rationale behind the assumed values of x and y, and couldn't any of the four propositions support it? And if x = y, how could I justify the argument that x and y were equal to two ostensibly different variables without showing that u = v? On one, hand I can sort of see that the assumption x = y and the conditions (1) - (4) would necessarily make it true that u = v...but, given that then either x or y could be said to be equal to either v or u, would there be any need for this part of the proof at all?
In the interest of completeness, the rest of the proof is:
Case 2: x ≠ y. In view of (1), either x = u or y = u. Suppose x ≠ u. Then y = u and by (3) x = v. On the other hand, suppose y ≠ u. Then x = u and by (4), y = v.
The theorem to be proved is
Theorem 44. {x,y} = {u,v} → (x = u & y = v) V (x = v & y = u)
where {x,y} and {u,v} are sets with exactly two members, which can be either sets or individuals. The proof relies on:
Theorem 43. z [itex]\in[/itex] {x,y} z = x V z = y.
The given proof is:
"By virtue of Theorem 43
u [itex]\in[/itex] {u,v},
and thus by the hypothesis of the theorem
u [itex]\in[/itex] {x,y}.
Hence, by virtue of Th. 43 again
(1)
u = x V u = y.
By exactly similar arguments
(2)
v = x V v = y,
(3)
x = u V x = v,
(4)
y = u V y = v.
We may now consider two cases.
Case 1: x = y. Then by virtue of (1), x = u, and by virtue of (2), y = v."
This is where I got lost. Couldn't I just as easily argue that x = v by virtue of (2) and y = u by virtue of (1)? Or by virtue of (3) and (4)? What's the rationale behind the assumed values of x and y, and couldn't any of the four propositions support it? And if x = y, how could I justify the argument that x and y were equal to two ostensibly different variables without showing that u = v? On one, hand I can sort of see that the assumption x = y and the conditions (1) - (4) would necessarily make it true that u = v...but, given that then either x or y could be said to be equal to either v or u, would there be any need for this part of the proof at all?
In the interest of completeness, the rest of the proof is:
Case 2: x ≠ y. In view of (1), either x = u or y = u. Suppose x ≠ u. Then y = u and by (3) x = v. On the other hand, suppose y ≠ u. Then x = u and by (4), y = v.
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