Theorem ssnelpss 3614

Description: A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)

Assertion

Ref	Expression
ssnelpss	|- (A C_ B -> ((C e. B /\ -. C e. A) -> A C. B))

Proof of Theorem ssnelpss

Step	Hyp	Ref	Expression
1		nelneq2 2452	. . 3 |- ((C e. B /\ -. C e. A) -> -. B = A)
2		eqcom 2355	. . 3 |- (B = A <-> A = B)
3	1, 2	sylnib 295	. 2 |- ((C e. B /\ -. C e. A) -> -. A = B)
4		dfpss2 3355	. . 3 |- (A C. B <-> (A C_ B /\ -. A = B))
5	4	baibr 872	. 2 |- (A C_ B -> (-. A = B <-> A C. B))
6	3, 5	syl5ib 210	1 |- (A C_ B -> ((C e. B /\ -. C e. A) -> A C. B))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   C_ wss 3258   C. wpss 3259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-ne 2519  df-pss 3262

This theorem is referenced by:  ssnelpssd  3615