Theorem dfpss2 3023

Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)

Assertion

Ref	Expression
dfpss2	|- (A C. B <-> (A C_ B /\ -. A = B))

Proof of Theorem dfpss2

Step	Hyp	Ref	Expression
1		df-pss 2927	. 2 |- (A C. B <-> (A C_ B /\ A =/= B))
2		df-ne 2203	. . 3 |- (A =/= B <-> -. A = B)
3	2	anbi2i 430	. 2 |- ((A C_ B /\ A =/= B) <-> (A C_ B /\ -. A = B))
4	1, 3	bitri 173	1 |- (A C. B <-> (A C_ B /\ -. A = B))

Syntax hints:  -. wn 3   /\ wa 97   <-> wb 98   = wceq 1242   =/= wne 2201   C_ wss 2911   C. wpss 2912

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-ne 2203  df-pss 2927

This theorem is referenced by:  dfpss3  3024  psstr  3043  sspsstr  3044  psssstr  3045  pssv  3261  disj4im  3270  ssnelpss  3283  onpsssuc  4247  f1imapss  5358