Theorem ssnelpss 3289

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 2139	. . 3 |- ( ( C e. B /\ -. C e. A ) -> -. B = A )
2		eqcom 2042	. . 3 |- ( B = A <-> A = B )
3	1, 2	sylnib 601	. 2 |- ( ( C e. B /\ -. C e. A ) -> -. A = B )
4		dfpss2 3029	. . 3 |- ( A C. B <-> ( A C_ B /\ -. A = B ) )
5	4	baibr 829	. 2 |- ( A C_ B -> ( -. A = B <-> A C. B ) )
6	3, 5	syl5ib 143	1 |- ( A C_ B -> ( ( C e. B /\ -. C e. A ) -> A C. B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   C_ wss 2917   C. wpss 2918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-ne 2206  df-pss 2933

This theorem is referenced by:  ssnelpssd  3290