Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - add the Axiom of Power Sets
Relations
relopabiv
Metamath Proof Explorer
Description: A class of ordered pairs is a relation. For a version without disjoint
variable condition, but a longer proof using ax-13 , see relopabi .
(Contributed by BJ , 22-Jul-2023)
Ref
Expression
Hypothesis
relopabiv.1
⊢ A = x y | φ
Assertion
relopabiv
⊢ Rel ⁡ A
Proof
Step
Hyp
Ref
Expression
1
relopabiv.1
⊢ A = x y | φ
2
vex
⊢ x ∈ V
3
vex
⊢ y ∈ V
4
2 3
pm3.2i
⊢ x ∈ V ∧ y ∈ V
5
4
a1i
⊢ φ → x ∈ V ∧ y ∈ V
6
5
ssopab2i
⊢ x y | φ ⊆ x y | x ∈ V ∧ y ∈ V
7
df-xp
⊢ V × V = x y | x ∈ V ∧ y ∈ V
8
6 1 7
3sstr4i
⊢ A ⊆ V × V
9
df-rel
⊢ Rel ⁡ A ↔ A ⊆ V × V
10
8 9
mpbir
⊢ Rel ⁡ A