Definition df-ssr 35898

Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5430) and identity relation (df-id 5425) classes. Subset relation class and Scott Fenton's subset class df-sset 33430 are the same: S = SSet (compare dfssr2 35899 with df-sset 33430), the only reason we do not use dfssr2 35899 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 3898) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 35901. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3898. The only exception (aside from directly investigating the class S e.g. in relssr 35900 or in extssr 35909) is when we have a specific purpose with its usage, like in case of df-refs 35910 versus df-cnvrefs 35923, 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 35728, extep 35700 and extssr 35909, 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 35616	. 2 class S
2		vx	. . . . 5 setvar 𝑥
3	2	cv 1537	. . . 4 class 𝑥
4		vy	. . . . 5 setvar 𝑦
5	4	cv 1537	. . . 4 class 𝑦
6	3, 5	wss 3881	. . 3 wff 𝑥 ⊆ 𝑦
7	6, 2, 4	copab 5092	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
8	1, 7	wceq 1538	1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}

This definition is referenced by:  dfssr2  35899  relssr  35900  brssr  35901