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Theorem ssnelpss 3613
 Description: A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
Assertion
Ref Expression
ssnelpss (A B → ((C B ¬ C A) → AB))

Proof of Theorem ssnelpss
StepHypRef Expression
1 nelneq2 2452 . . 3 ((C B ¬ C A) → ¬ B = A)
2 eqcom 2355 . . 3 (B = AA = B)
31, 2sylnib 295 . 2 ((C B ¬ C A) → ¬ A = B)
4 dfpss2 3354 . . 3 (AB ↔ (A B ¬ A = B))
54baibr 872 . 2 (A B → (¬ A = BAB))
63, 5syl5ib 210 1 (A B → ((C B ¬ C A) → AB))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257   ⊊ wpss 3258 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2518  df-pss 3261 This theorem is referenced by:  ssnelpssd  3614
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