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Definition df-ssr 36379
Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5474) and identity relation (df-id 5469) classes. Subset relation class and Scott Fenton's subset class df-sset 33921 are the same: S = SSet (compare dfssr2 36380 with df-sset 33921), the only reason we do not use dfssr2 36380 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 3897) are the same, that is, (𝐴 S 𝐵𝐴𝐵) when 𝐵 is a set, see brssr 36382. Yet in general we use the subclass relation 𝐴𝐵 both for classes and for sets, see the comment of df-ss 3897. The only exception (aside from directly investigating the class S e.g. in relssr 36381 or in extssr 36390) is when we have a specific purpose with its usage, like in case of df-refs 36391 versus df-cnvrefs 36404, 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 36209, extep 36181 and extssr 36390, then "extrelssr" " |- ExtRel S " is a theorem along with "extrelep" " |- ExtRel E " and "extrelid" " |- ExtRel I " . (Contributed by Peter Mazsa, 25-Jul-2019.)

Assertion
Ref Expression
df-ssr S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-ssr
StepHypRef Expression
1 cssr 36099 . 2 class S
2 vx . . . . 5 setvar 𝑥
32cv 1542 . . . 4 class 𝑥
4 vy . . . . 5 setvar 𝑦
54cv 1542 . . . 4 class 𝑦
63, 5wss 3880 . . 3 wff 𝑥𝑦
76, 2, 4copab 5129 . 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
81, 7wceq 1543 1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
Colors of variables: wff setvar class
This definition is referenced by:  dfssr2  36380  relssr  36381  brssr  36382
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