Definition df-ssr 38489

Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5538) and identity relation (df-id 5533) classes. Subset relation class and Scott Fenton's subset class df-sset 35844 are the same: S = SSet (compare dfssr2 38490 with df-sset 35844), the only reason we do not use dfssr2 38490 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 3931) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 38492. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3931. The only exception (aside from directly investigating the class S e.g. in relssr 38491 or in extssr 38500) is when we have a specific purpose with its usage, like in case of df-refs 38501 versus df-cnvrefs 38516, 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 38298, extep 38271 and extssr 38500, 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 38172	. 2 class S
2		vx	. . . . 5 setvar 𝑥
3	2	cv 1539	. . . 4 class 𝑥
4		vy	. . . . 5 setvar 𝑦
5	4	cv 1539	. . . 4 class 𝑦
6	3, 5	wss 3914	. . 3 wff 𝑥 ⊆ 𝑦
7	6, 2, 4	copab 5169	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
8	1, 7	wceq 1540	1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}

This definition is referenced by:  dfssr2  38490  relssr  38491  brssr  38492