Definition df-pss 2927

Description: Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. Note that ¬ A ⊊ A (proved in pssirr 3038). Contrast this relationship with the relationship A ⊆ B (as defined in df-ss 2925). Other possible definitions are given by dfpss2 3023 and dfpss3 3024. (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 2912	. 2 wff A ⊊ B
4	1, 2	wss 2911	. . 3 wff A ⊆ B
5	1, 2	wne 2201	. . 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  3023  psseq1  3025  psseq2  3026  pssss  3033  pssne  3034  nssinpss  3163  0pss  3259  difsnpssim  3498