Theorem vss 3456

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

Step	Hyp	Ref	Expression
1		ssv 3164	. . 3 |- A C_ _V
2	1	biantrur 301	. 2 |- ( _V C_ A <-> ( A C_ _V /\ _V C_ A ) )
3		eqss 3157	. 2 |- ( A = _V <-> ( A C_ _V /\ _V C_ A ) )
4	2, 3	bitr4i 186	1 |- ( _V C_ A <-> A = _V )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   _Vcvv 2726   C_ wss 3116

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728  df-in 3122  df-ss 3129

This theorem is referenced by:  vdif0im  3474