Definition df-ssr 37363

Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5580) and identity relation (df-id 5574) classes. Subset relation class and Scott Fenton's subset class df-sset 34823 are the same: S = SSet (compare dfssr2 37364 with df-sset 34823), the only reason we do not use dfssr2 37364 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 3965) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 37366. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3965. The only exception (aside from directly investigating the class S e.g. in relssr 37365 or in extssr 37374) is when we have a specific purpose with its usage, like in case of df-refs 37375 versus df-cnvrefs 37390, 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 37174, extep 37146 and extssr 37374, 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 37041	. 2 class S
2		vx	. . . . 5 setvar 𝑥
3	2	cv 1540	. . . 4 class 𝑥
4		vy	. . . . 5 setvar 𝑦
5	4	cv 1540	. . . 4 class 𝑦
6	3, 5	wss 3948	. . 3 wff 𝑥 ⊆ 𝑦
7	6, 2, 4	copab 5210	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
8	1, 7	wceq 1541	1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}

This definition is referenced by:  dfssr2  37364  relssr  37365  brssr  37366