Theorem ssnelpss 2937

Description: A subclass missing a member is a proper subclass.

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		dfpss2 2694	. . 3 |- (A C. B <-> (A C_ B /\ -. A = B))
2	1	baibr 750	. 2 |- (A C_ B -> (-. A = B <-> A C. B))
3		nelneq2 1986	. . 3 |- ((C e. B /\ -. C e. A) -> -. B = A)
4		eqcom 1886	. . . 4 |- (B = A <-> A = B)
5	4	notbii 204	. . 3 |- (-. B = A <-> -. A = B)
6	3, 5	sylib 215	. 2 |- ((C e. B /\ -. C e. A) -> -. A = B)
7	2, 6	syl5bi 225	1 |- (A C_ B -> ((C e. B /\ -. C e. A) -> A C. B))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593   C. wpss 2594

This theorem is referenced by:  nthruc 7995  nthruz 7996

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-cleq 1877  df-clel 1880  df-ne 2019  df-pss 2607

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