Definition df-ssr 35742

Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5468) and identity relation (df-id 5463) classes. Subset relation class and Scott Fenton's subset class df-sset 33321 are the same: S = SSet (compare dfssr2 35743 with df-sset 33321), the only reason we do not use dfssr2 35743 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 3955) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 35745. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3955. The only exception (aside from directly investigating the class S e.g. in relssr 35744 or in extssr 35753) is when we have a specific purpose with its usage, like in case of df-refs 35754 versus df-cnvrefs 35767, 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 35572, extep 35544 and extssr 35753, 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 35460	. 2 class S
2		vx	. . . . 5 setvar 𝑥
3	2	cv 1535	. . . 4 class 𝑥
4		vy	. . . . 5 setvar 𝑦
5	4	cv 1535	. . . 4 class 𝑦
6	3, 5	wss 3939	. . 3 wff 𝑥 ⊆ 𝑦
7	6, 2, 4	copab 5131	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
8	1, 7	wceq 1536	1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}

This definition is referenced by:  dfssr2  35743  relssr  35744  brssr  35745