MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssnelpss Structured version   Visualization version   GIF version

Theorem ssnelpss 3939
Description: A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
Assertion
Ref Expression
ssnelpss (𝐴𝐵 → ((𝐶𝐵 ∧ ¬ 𝐶𝐴) → 𝐴𝐵))

Proof of Theorem ssnelpss
StepHypRef Expression
1 nelneq2 2883 . . 3 ((𝐶𝐵 ∧ ¬ 𝐶𝐴) → ¬ 𝐵 = 𝐴)
2 eqcom 2784 . . 3 (𝐵 = 𝐴𝐴 = 𝐵)
31, 2sylnib 320 . 2 ((𝐶𝐵 ∧ ¬ 𝐶𝐴) → ¬ 𝐴 = 𝐵)
4 dfpss2 3913 . . 3 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))
54baibr 532 . 2 (𝐴𝐵 → (¬ 𝐴 = 𝐵𝐴𝐵))
63, 5syl5ib 236 1 (𝐴𝐵 → ((𝐶𝐵 ∧ ¬ 𝐶𝐴) → 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1601  wcel 2106  wss 3791  wpss 3792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-cleq 2769  df-clel 2773  df-ne 2969  df-pss 3807
This theorem is referenced by:  ssnelpssd  3940  ssexnelpss  3941  canthp1lem2  9810  nqpr  10171  uzindi  13100  nthruc  15385  nthruz  15386  vitali  23817  onpsstopbas  33012
  Copyright terms: Public domain W3C validator