Definition df-ss 3259

Description: Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For example, { 1 , 2 } ⊆ { 1 , 2 , 3 } (ex-ss in set.mm). Note that A ⊆ A (proved in ssid 3290). Contrast this relationship with the relationship A ⊊ B (as will be defined in df-pss 3261). For a more traditional definition, but requiring a dummy variable, see dfss2 3262. Other possible definitions are given by dfss3 3263, dfss4 3489, sspss 3368, ssequn1 3433, ssequn2 3436, sseqin2 3474, and ssdif0 3609. (Contributed by NM, 27-Apr-1994.)

Assertion

Ref	Expression
df-ss	⊢ (A ⊆ B ↔ (A ∩ B) = A)

Detailed syntax breakdown of Definition df-ss

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cB	. . 3 class B
3	1, 2	wss 3257	. 2 wff A ⊆ B
4	1, 2	cin 3208	. . 3 class (A ∩ B)
5	4, 1	wceq 1642	. 2 wff (A ∩ B) = A
6	3, 5	wb 176	1 wff (A ⊆ B ↔ (A ∩ B) = A)

This definition is referenced by:  dfss  3260  dfss1  3460  inabs  3486  nssinpss  3487  dfrab3ss  3533  disjssun  3608  riinn0  4040  rintn0  4056  dfiota4  4372  ssfin  4470  ssdmres  4987  resabs1  4992