Definition df-ssr 34876

Description: Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5266) and identity relation (df-id 5261) classes. Subset relation class and Scott Fenton's subset class df-sset 32552 are the same: S = SSet (compare dfssr2 34877 with df-sset 32552, cf. comment of df-xrn 34761), the only reason we do not use dfssr2 34877 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 3806) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, cf. brssr 34879. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, cf. the comment of df-ss 3806. The only exception (aside from directly investigating the class S e.g. in relssr 34878 or in extssr 34887) is when we have a specific purpose with its usage, like in case of df-refs 34888 versus df-cnvrefs 34901, 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 34710, extep 34680 and extssr 34887, 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 34609	. 2 class S
2		vx	. . . . 5 setvar 𝑥
3	2	cv 1600	. . . 4 class 𝑥
4		vy	. . . . 5 setvar 𝑦
5	4	cv 1600	. . . 4 class 𝑦
6	3, 5	wss 3792	. . 3 wff 𝑥 ⊆ 𝑦
7	6, 2, 4	copab 4948	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}
8	1, 7	wceq 1601	1 wff S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦}

This definition is referenced by:  dfssr2  34877  relssr  34878  brssr  34879