We define equality as follows:
a
=
b
:
⟺
d
e
f
i
n
i
t
i
o
n
∀
x
:
x
∈
a
⇔
x
∈
b
{\displaystyle a=b:{\stackrel {\mathrm {definition} }{\Longleftrightarrow }}\ \forall x:\ x\in a\Leftrightarrow x\in b}
It follows:
Reflexivity.
a
=
a
{\displaystyle a=a}
∀
x
x
∈
a
⇔
x
∈
a
{\displaystyle \forall x\ x\in a\Leftrightarrow x\in a}
, which always holds
a
=
b
⇒
b
=
a
{\displaystyle a=b\Rightarrow b=a}
a
=
b
∧
b
=
c
⇒
a
=
c
{\displaystyle a=b\land b=c\Rightarrow a=c}
Extensionality, two sets with the same elements are equal.
∀
x
,
y
,
z
(
z
∈
x
⇔
z
∈
y
)
⇒
(
x
=
y
)
{\displaystyle \forall x,y,z\ (z\in x\Leftrightarrow z\in y)\Rightarrow (x=y)}
Separation, subsets exist
∀
y
1
,
p
∃
y
2
∀
x
x
∈
y
2
⇔
(
p
∧
x
∈
y
1
)
{\displaystyle \forall y_{1},p\ \exists y_{2}\ \forall x\ x\in y_{2}\Leftrightarrow (p\wedge x\in y_{1})}
where p is any proposition
∃
x
∀
y
y
∉
x
{\displaystyle \exists x\ \forall y\ y\not \in x}
Union, the union of all members of a set is a set.
∀
x
∃
y
∀
z
z
∈
y
⇔
(
∃
u
z
∈
u
∧
u
∈
x
)
{\displaystyle \forall x\ \exists y\ \forall z\ z\in y\Leftrightarrow (\exists u\ z\in u\wedge u\in x)}
∀
x
∃
y
∀
z
z
∈
y
⇔
(
∀
t
t
∈
z
⇒
t
∈
x
)
{\displaystyle \forall x\ \exists y\ \forall z\ z\in y\Leftrightarrow (\forall t\ t\in z\Rightarrow t\in x)}
we denote this set y by P(x)
Infinity, an infinite set exists
∃
x
(
∅
∈
x
)
∧
(
∀
y
y
∈
x
⇒
P
(
y
)
∈
x
)
{\displaystyle \exists x\ (\emptyset \in x)\wedge (\forall y\ y\in x\Rightarrow P(y)\in x)}
Foundation, no set is a member of itself
∀
x
x
≠
∅
⇒
(
∃
y
∈
x
y
∩
x
=
∅
)
{\displaystyle \forall x\ x\neq \emptyset \Rightarrow (\exists y\in x\ y\cap x=\emptyset )}
← Cardinals · Naive Set Theory →