Definition df-ssr 39478

Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5551) and identity relation (df-id 5546) classes. Subset relation class and Scott Fenton's subset class df-sset 36588 are the same: S = SSet (compare dfssr2 39479 with df-sset 36588), the only reason we do not use dfssr2 39479 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 3916) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 39481. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3916. The only exception (aside from directly investigating the class S e.g. in relssr 39480 or in extssr 39489) is when we have a specific purpose with its usage, like in case of df-refs 39490 versus df-cnvrefs 39505, 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 39216, extep 39189 and extssr 39489, 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 39086	. 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 3899	. . 3 wff 𝑥 ⊆ 𝑦
7	6, 2, 4	copab 5167	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
8	1, 7	wceq 1570	1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}

This definition is used by:  dfssr2  39479  relssr  39480  brssr  39481