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Theorem vss 4447
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 4006 . . 3 𝐴 ⊆ V
21biantrur 529 . 2 (V ⊆ 𝐴 ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
3 eqss 3997 . 2 (𝐴 = V ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
42, 3bitr4i 277 1 (V ⊆ 𝐴𝐴 = V)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  Vcvv 3473  wss 3949
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-v 3475  df-in 3956  df-ss 3966
This theorem is referenced by:  vdif0  4472
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