Theorem ssnelpss 3718

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 2726	. . 3 |- ( ( C e. B /\ -. C e. A ) -> -. B = A )
2		eqcom 2629	. . 3 |- ( B = A <-> A = B )
3	1, 2	sylnib 318	. 2 |- ( ( C e. B /\ -. C e. A ) -> -. A = B )
4		dfpss2 3692	. . 3 |- ( A C. B <-> ( A C_ B /\ -. A = B ) )
5	4	baibr 945	. 2 |- ( A C_ B -> ( -. A = B <-> A C. B ) )
6	3, 5	syl5ib 234	1 |- ( A C_ B -> ( ( C e. B /\ -. C e. A ) -> A C. B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   C. wpss 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-ne 2795  df-pss 3590

This theorem is referenced by:  ssnelpssd  3719  ssexnelpss  3720  canthp1lem2  9475  nqpr  9836  uzindi  12781  nthruc  14981  nthruz  14982  vitali  23382  onpsstopbas  32429