Ordered pair
Proving that definitions satisfy the characteristic property

Prove: (a, b) = (c, d) if and only if a = c and b = d.

Kuratowski: If. If a = c and b = d, then {{ a }, { a, b }} = {{ c }, { c, d }}. Thus (a, b) = (c, d).

Only if. Two cases: a = b, and a ≠ b.

If a = b:

If a ≠ b, then (a, b) = (c, d) implies {{ a }, { a, b }} = {{ c }, { c, d }}.

Reverse: (a, b) = {{ b }, { a, b }} = {{ b }, { b, a }} = (b, a).

If. If (a, b) = (c, d),(b, a) = (d, c). Therefore, b = d and a = c.

Only if. If a = c and b = d, then {{ b }, { a, b }} = {{ d }, { c, d }}.Thus (a, b) = (c, d).

Short:

If: If a = c and b = d, then { a, { a, b }} = { c, { c, d }}. Thus (a, b) = (c, d).

Only if: Suppose { a, { a, b }} = { c, { c, d }}.Then a is in the left hand side, and thus in the right hand side.Because equal sets have equal elements, one of a = c or a = { c, d } must be the case.

Again, we see that { a, b } = c or { a, b } = { c, d }.
keywords:forbidden_words, length_constraints:number_sentences[8]