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Theorem pssv 4442
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 4002 . 2 𝐴 ⊆ V
2 dfpss2 4081 . 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 205   = wceq 1534  Vcvv 3470  wss 3945  wpss 3946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-v 3472  df-in 3952  df-ss 3962  df-pss 3964
This theorem is referenced by: (None)
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