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Theorem vss 4208
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 3821 . . 3 𝐴 ⊆ V
21biantrur 527 . 2 (V ⊆ 𝐴 ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
3 eqss 3813 . 2 (𝐴 = V ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
42, 3bitr4i 270 1 (V ⊆ 𝐴𝐴 = V)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  Vcvv 3385  wss 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-v 3387  df-in 3776  df-ss 3783
This theorem is referenced by:  vdif0  4231
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