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Theorem vss 3292
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  C_  A  <->  A  =  _V )

Proof of Theorem vss
StepHypRef Expression
1 ssv 3020 . . 3  |-  A  C_  _V
21biantrur 297 . 2  |-  ( _V  C_  A  <->  ( A  C_  _V  /\  _V  C_  A
) )
3 eqss 3015 . 2  |-  ( A  =  _V  <->  ( A  C_ 
_V  /\  _V  C_  A
) )
42, 3bitr4i 185 1  |-  ( _V  C_  A  <->  A  =  _V )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   _Vcvv 2602    C_ wss 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604  df-in 2980  df-ss 2987
This theorem is referenced by:  vdif0im  3310
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