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Theorem pssv 3968
 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 3588 . 2 𝐴 ⊆ V
2 dfpss2 3654 . 2 (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V))
31, 2mpbiran 955 1 (𝐴 ⊊ V ↔ ¬ 𝐴 = V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475  Vcvv 3173   ⊆ wss 3540   ⊊ wpss 3541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ne 2782  df-v 3175  df-in 3547  df-ss 3554  df-pss 3556 This theorem is referenced by: (None)
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