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Theorem vss 3441
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 3150 . . 3  |-  A  C_  _V
21biantrur 301 . 2  |-  ( _V  C_  A  <->  ( A  C_  _V  /\  _V  C_  A
) )
3 eqss 3143 . 2  |-  ( A  =  _V  <->  ( A  C_ 
_V  /\  _V  C_  A
) )
42, 3bitr4i 186 1  |-  ( _V  C_  A  <->  A  =  _V )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1335   _Vcvv 2712    C_ wss 3102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714  df-in 3108  df-ss 3115
This theorem is referenced by:  vdif0im  3459
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