**Description: **Define the subsets class
or the class of subset relations. Similar to
definitions of epsilon relation (df-eprel 5266) and identity relation
(df-id 5261) classes. Subset relation class and Scott
Fenton's subset
class df-sset 32552 are the same: S = SSet (compare dfssr2 34877 with
df-sset 32552, cf. comment of df-xrn 34761), the only reason we do not use
dfssr2 34877 as the base definition of the subsets class
is the way we
defined the epsilon relation and the identity relation classes.
The binary relation on the class of subsets and the subclass
relationship (df-ss 3806) are the same, that is,
(𝐴
S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, cf. brssr 34879. Yet in
general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for
sets, cf. the comment of df-ss 3806. The only exception (aside from
directly investigating the class S e.g. in relssr 34878 or in
extssr 34887) is when we have a specific purpose with its
usage, like in
case of df-refs 34888 versus df-cnvrefs 34901, where we need S to
define
the class of reflexive sets in order to be able to define the class of
converse reflexive sets with the help of the converse of S.
The subsets class S has another place in set.mm
as well: if we
define extensional relation based on the common property in extid 34710,
extep 34680 and extssr 34887, then "extrelssr" " |- ExtRel
_S " is a theorem
along with "extrelep" " |- ExtRel _E " and
"extrelid" " |- ExtRel _I ".
(Contributed by Peter Mazsa, 25-Jul-2019.) |