Definition df-pss 2930

Description: Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. Note that ¬ 𝐴 ⊊ 𝐴 (proved in pssirr 3041). Contrast this relationship with the relationship 𝐴 ⊆ 𝐵 (as defined in df-ss 2928). Other possible definitions are given by dfpss2 3026 and dfpss3 3027. (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 2915	. 2 wff 𝐴 ⊊ 𝐵
4	1, 2	wss 2914	. . 3 wff 𝐴 ⊆ 𝐵
5	1, 2	wne 2204	. . 3 wff 𝐴 ≠ 𝐵
6	4, 5	wa 97	. 2 wff (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)
7	3, 6	wb 98	1 wff (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))

This definition is referenced by:  dfpss2  3026  psseq1  3028  psseq2  3029  pssss  3036  pssne  3037  nssinpss  3166  0pss  3262  difsnpssim  3504