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Disjoints vs. converse functions
eldisjs3
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eldisjs4
Metamath Proof Explorer
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Theorem
eldisjs3
Description:
Elementhood in the class of disjoints.
(Contributed by
Peter Mazsa
, 5-Sep-2021)
Ref
Expression
Assertion
eldisjs3
⊢
R
∈
Disjs
↔
∀
u
∀
v
∀
x
u
R
x
∧
v
R
x
→
u
=
v
∧
R
∈
Rels
Proof
Step
Hyp
Ref
Expression
1
eldisjs2
⊢
R
∈
Disjs
↔
≀
R
-1
⊆
I
∧
R
∈
Rels
2
cosscnvssid3
⊢
≀
R
-1
⊆
I
↔
∀
u
∀
v
∀
x
u
R
x
∧
v
R
x
→
u
=
v
3
2
anbi1i
⊢
≀
R
-1
⊆
I
∧
R
∈
Rels
↔
∀
u
∀
v
∀
x
u
R
x
∧
v
R
x
→
u
=
v
∧
R
∈
Rels
4
1
3
bitri
⊢
R
∈
Disjs
↔
∀
u
∀
v
∀
x
u
R
x
∧
v
R
x
→
u
=
v
∧
R
∈
Rels