User:Dfeuer/Equality of Ordered Pairs implies Equality of Elements/Lemma 2

Theorem

Let $a$, $b$, and $c$ be sets.

Then:

$\{a, b\} = \{a, c\} \implies b = c$

Proof

Suppose that $\{a, b\} = \{a, c\}$.

By the definition of unordered pair:

$b \in \{a, b\}$ and

$c \in \{a, c\}$

By assumption, then:

$b \in \{a, c\}$ and

$c \in \{a, b\}$

By the definition of unordered pair:

$b = a \lor b = c$ and

$c = a \lor b = c$

Since disjunction distributes over conjunction:

$(b = a \land c = a) \lor b = c$.

If $b = a \land c = a$ then $b = c$.

Thus in either case, $b = c$.

$\blacksquare$

Sources

SF Lemma 2.4.3