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Theorem ssnelpss 3266
 Description: A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
Assertion
Ref Expression
ssnelpss (AB → ((𝐶 B ¬ 𝐶 A) → AB))

Proof of Theorem ssnelpss
StepHypRef Expression
1 nelneq2 2121 . . 3 ((𝐶 B ¬ 𝐶 A) → ¬ B = A)
2 eqcom 2024 . . 3 (B = AA = B)
31, 2sylnib 588 . 2 ((𝐶 B ¬ 𝐶 A) → ¬ A = B)
4 dfpss2 3006 . . 3 (AB ↔ (AB ¬ A = B))
54baibr 819 . 2 (AB → (¬ A = BAB))
63, 5syl5ib 143 1 (AB → ((𝐶 B ¬ 𝐶 A) → AB))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374   ⊆ wss 2894   ⊊ wpss 2895 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 532  ax-in2 533  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-cleq 2015  df-clel 2018  df-ne 2188  df-pss 2910 This theorem is referenced by:  ssnelpssd  3267
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