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

Theorem vss 3964
Description: Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
vss (V ⊆ 𝐴𝐴 = V)

Proof of Theorem vss
StepHypRef Expression
1 ssv 3588 . . 3 𝐴 ⊆ V
21biantrur 526 . 2 (V ⊆ 𝐴 ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
3 eqss 3583 . 2 (𝐴 = V ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
42, 3bitr4i 266 1 (V ⊆ 𝐴𝐴 = V)
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  Vcvv 3173  wss 3540
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-v 3175  df-in 3547  df-ss 3554
This theorem is referenced by:  vdif0  3989
  Copyright terms: Public domain W3C validator