Definition df-ssr 38823

Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5532) and identity relation (df-id 5527) classes. Subset relation class and Scott Fenton's subset class df-sset 36067 are the same: S = SSet (compare dfssr2 38824 with df-sset 36067), the only reason we do not use dfssr2 38824 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 3920) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 38826. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3920. The only exception (aside from directly investigating the class S e.g. in relssr 38825 or in extssr 38834) is when we have a specific purpose with its usage, like in case of df-refs 38835 versus df-cnvrefs 38850, 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 38561, extep 38534 and extssr 38834, 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 38431	. 2 class S
2		vx	. . . . 5 setvar 𝑥
3	2	cv 1541	. . . 4 class 𝑥
4		vy	. . . . 5 setvar 𝑦
5	4	cv 1541	. . . 4 class 𝑦
6	3, 5	wss 3903	. . 3 wff 𝑥 ⊆ 𝑦
7	6, 2, 4	copab 5162	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
8	1, 7	wceq 1542	1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}

This definition is referenced by:  dfssr2  38824  relssr  38825  brssr  38826