Definition df-pss 3262

Description: Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. For example, { 1 , 2 } ⊊ { 1 , 2 , 3 } (ex-pss in set.mm). Note that ¬ A ⊊ A (proved in pssirr 3370). Contrast this relationship with the relationship A ⊆ B (as defined in df-ss 3260). Other possible definitions are given by dfpss2 3355 and dfpss3 3356. (Contributed by NM, 7-Feb-1996.)

Assertion

Ref	Expression
df-pss	⊢ (A ⊊ B ↔ (A ⊆ B ∧ A ≠ B))

Detailed syntax breakdown of Definition df-pss

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cB	. . 3 class B
3	1, 2	wpss 3259	. 2 wff A ⊊ B
4	1, 2	wss 3258	. . 3 wff A ⊆ B
5	1, 2	wne 2517	. . 3 wff A ≠ B
6	4, 5	wa 358	. 2 wff (A ⊆ B ∧ A ≠ B)
7	3, 6	wb 176	1 wff (A ⊊ B ↔ (A ⊆ B ∧ A ≠ B))

This definition is used by:  dfpss2  3355  psseq1  3357  psseq2  3358  pssss  3365  pssne  3366  nssinpss  3488  0pss  3589  pssdif  3613  difsnpss  3852  sfinltfin  4536