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Theorem vss 3264
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 2965 . . 3  |-  A  C_  _V
21biantrur 287 . 2  |-  ( _V  C_  A  <->  ( A  C_  _V  /\  _V  C_  A
) )
3 eqss 2960 . 2  |-  ( A  =  _V  <->  ( A  C_ 
_V  /\  _V  C_  A
) )
42, 3bitr4i 176 1  |-  ( _V  C_  A  <->  A  =  _V )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98    = wceq 1243   _Vcvv 2557    C_ wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559  df-in 2924  df-ss 2931
This theorem is referenced by:  vdif0im  3287
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