Definition df-ssr 38610

Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5519) and identity relation (df-id 5514) classes. Subset relation class and Scott Fenton's subset class df-sset 35919 are the same: S = SSet (compare dfssr2 38611 with df-sset 35919), the only reason we do not use dfssr2 38611 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 3915) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 38613. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3915. The only exception (aside from directly investigating the class S e.g. in relssr 38612 or in extssr 38621) is when we have a specific purpose with its usage, like in case of df-refs 38622 versus df-cnvrefs 38637, 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 38368, extep 38341 and extssr 38621, 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 38245	. 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 3898	. . 3 wff 𝑥 ⊆ 𝑦
7	6, 2, 4	copab 5155	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
8	1, 7	wceq 1541	1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}

This definition is referenced by:  dfssr2  38611  relssr  38612  brssr  38613