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SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Peter Mazsa
Symmetry
elsymrels2
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elsymrels3
Metamath Proof Explorer
Ascii
Unicode
Theorem
elsymrels2
Description:
Element of the class of symmetric relations.
(Contributed by
Peter Mazsa
, 17-Aug-2021)
Ref
Expression
Assertion
elsymrels2
⊢
R
∈
SymRels
↔
R
-1
⊆
R
∧
R
∈
Rels
Proof
Step
Hyp
Ref
Expression
1
dfsymrels2
⊢
SymRels
=
r
∈
Rels
|
r
-1
⊆
r
2
cnveq
⊢
r
=
R
→
r
-1
=
R
-1
3
id
⊢
r
=
R
→
r
=
R
4
2
3
sseq12d
⊢
r
=
R
→
r
-1
⊆
r
↔
R
-1
⊆
R
5
1
4
rabeqel
⊢
R
∈
SymRels
↔
R
-1
⊆
R
∧
R
∈
Rels