Definition df-ssr 39268

Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5566) and identity relation (df-id 5561) classes. Subset relation class and Scott Fenton's subset class df-sset 36367 are the same: S = SSet (compare dfssr2 39269 with df-sset 36367), the only reason we do not use dfssr2 39269 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 3925) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 39271. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3925. The only exception (aside from directly investigating the class S e.g. in relssr 39270 or in extssr 39279) is when we have a specific purpose with its usage, like in case of df-refs 39280 versus df-cnvrefs 39295, 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 39006, extep 38979 and extssr 39279, 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 38876	. 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 3908	. . 3 wff 𝑥 ⊆ 𝑦
7	6, 2, 4	copab 5178	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
8	1, 7	wceq 1570	1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}

This definition is used by:  dfssr2  39269  relssr  39270  brssr  39271