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Theorem pssv 4381
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 3945 . 2 𝐴 ⊆ V
2 dfpss2 4020 . 2 (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V))
31, 2mpbiran 706 1 (𝐴 ⊊ V ↔ ¬ 𝐴 = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1539  Vcvv 3430  wss 3887  wpss 3888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3432  df-in 3894  df-ss 3904  df-pss 3906
This theorem is referenced by: (None)
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