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

Theorem pssv 4450
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 4006 . 2 𝐴 ⊆ V
2 dfpss2 4085 . 2 (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V))
31, 2mpbiran 707 1 (𝐴 ⊊ V ↔ ¬ 𝐴 = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1533  Vcvv 3473  wss 3949  wpss 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-v 3475  df-in 3956  df-ss 3966  df-pss 3968
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator