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Theorem ssnelpss 3283
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 2136 . . 3 ((𝐶 B ¬ 𝐶 A) → ¬ B = A)
2 eqcom 2039 . . 3 (B = AA = B)
31, 2sylnib 600 . 2 ((𝐶 B ¬ 𝐶 A) → ¬ A = B)
4 dfpss2 3023 . . 3 (AB ↔ (AB ¬ A = B))
54baibr 828 . 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 1242   wcel 1390  wss 2911  wpss 2912
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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033  df-ne 2203  df-pss 2927
This theorem is referenced by:  ssnelpssd  3284
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