Definition df-ssr 39713

Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5556) and identity relation (df-id 5551) classes. Subset relation class and Scott Fenton's subset class df-sset 36809 are the same: S = SSet (compare dfssr2 39714 with df-sset 36809), the only reason we do not use dfssr2 39714 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 3901) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 39716. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3901. The only exception (aside from directly investigating the class S e.g. in relssr 39715 or in extssr 39724) is when we have a specific purpose with its usage, like in case of df-refs 39725 versus df-cnvrefs 39740, 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 39451, extep 39424 and extssr 39724, 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 39321	. 2 class S
2		vx	. . . . 5 setvar 𝑥
3	2	cv 1569	. . . 4 class 𝑥
4		vy	. . . . 5 setvar 𝑦
5	4	cv 1569	. . . 4 class 𝑦
6	3, 5	wss 3900	. . 3 wff 𝑥 ⊆ 𝑦
7	6, 2, 4	copab 5171	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
8	1, 7	wceq 1570	1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}

This definition is used by:  dfssr2  39714  relssr  39715  brssr  39716