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Theorem pssv 4370
 Description: Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
pssv (𝐴 ⊊ V ↔ ¬ 𝐴 = V)

Proof of Theorem pssv
StepHypRef Expression
1 ssv 3966 . 2 𝐴 ⊆ V
2 dfpss2 4037 . 2 (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V))
31, 2mpbiran 708 1 (𝐴 ⊊ V ↔ ¬ 𝐴 = V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538  Vcvv 3469   ⊆ wss 3908   ⊊ wpss 3909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ne 3012  df-v 3471  df-in 3915  df-ss 3925  df-pss 3927 This theorem is referenced by: (None)
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