Theorem ssnelpss 1751

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

Assertion

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

Proof of Theorem ssnelpss

Step	Hyp	Ref	Expression
1		dfpss2 1557	. . 3 |- (A (. B <-> (A (_ B /\ -. A = B))
2	1	baibr 507	. 2 |- (A (_ B -> (-. A = B <-> A (. B))
3		clneq2 1169	. . 3 |- ((C e. B /\ -. C e. A) -> -. B = A)
4		cleqcom 1103	. . . 4 |- (B = A <-> A = B)
5	4	negbii 162	. . 3 |- (-. B = A <-> -. A = B)
6	3, 5	sylib 173	. 2 |- ((C e. B /\ -. C e. A) -> -. A = B)
7	2, 6	syl5bi 183	1 |- (A (_ B -> ((C e. B /\ -. C e. A) -> A (. B))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092   (_ wss 1487   (. wpss 1488

This theorem is referenced by:  nthruc 4784  nthruz 4785

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-ne 1192  df-pss 1494

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