Definition df-ss 3260

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 3291). Contrast this relationship with the relationship A ⊊ B (as will be defined in df-pss 3262). For a more traditional definition, but requiring a dummy variable, see dfss2 3263. Other possible definitions are given by dfss3 3264, dfss4 3490, sspss 3369, ssequn1 3434, ssequn2 3437, sseqin2 3475, and ssdif0 3610. (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 3258	. 2 wff A ⊆ B
4	1, 2	cin 3209	. . 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  3261  dfss1  3461  inabs  3487  nssinpss  3488  dfrab3ss  3534  disjssun  3609  riinn0  4041  rintn0  4057  dfiota4  4373  ssfin  4471  ssdmres  4988  resabs1  4993