Theorem dfpss2 3355

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 3262	. 2 |- (A C. B <-> (A C_ B /\ A =/= B))
2		df-ne 2519	. . 3 |- (A =/= B <-> -. A = B)
3	2	anbi2i 675	. 2 |- ((A C_ B /\ A =/= B) <-> (A C_ B /\ -. A = B))
4	1, 3	bitri 240	1 |- (A C. B <-> (A C_ B /\ -. A = B))

Syntax hints:  -. wn 3   <-> wb 176   /\ wa 358   = wceq 1642   =/= wne 2517   C_ wss 3258   C. wpss 3259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-ne 2519  df-pss 3262

This theorem is referenced by:  dfpss3  3356  sspss  3369  psstr  3374  npss  3380  pssv  3591  disj4  3600  ssnelpss  3614  sfinltfin  4536