Definition df-ssr 36543

Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5486) and identity relation (df-id 5480) classes. Subset relation class and Scott Fenton's subset class df-sset 34085 are the same: S = SSet (compare dfssr2 36544 with df-sset 34085), the only reason we do not use dfssr2 36544 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 3900) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 36546. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3900. The only exception (aside from directly investigating the class S e.g. in relssr 36545 or in extssr 36554) is when we have a specific purpose with its usage, like in case of df-refs 36555 versus df-cnvrefs 36568, 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 36373, extep 36345 and extssr 36554, 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⟩ ∣ 𝑥 ⊆ 𝑦}

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-ssr

Step	Hyp	Ref	Expression
1		cssr 36263	. 2 class S
2		vx	. . . . 5 setvar 𝑥
3	2	cv 1538	. . . 4 class 𝑥
4		vy	. . . . 5 setvar 𝑦
5	4	cv 1538	. . . 4 class 𝑦
6	3, 5	wss 3883	. . 3 wff 𝑥 ⊆ 𝑦
7	6, 2, 4	copab 5132	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
8	1, 7	wceq 1539	1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}

This definition is referenced by:  dfssr2  36544  relssr  36545  brssr  36546