Theorem vss 3556

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 3260	. . 3 |- A C_ _V
2	1	biantrur 303	. 2 |- ( _V C_ A <-> ( A C_ _V /\ _V C_ A ) )
3		eqss 3253	. 2 |- ( A = _V <-> ( A C_ _V /\ _V C_ A ) )
4	2, 3	bitr4i 187	1 |- ( _V C_ A <-> A = _V )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   _Vcvv 2813   C_ wss 3211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-v 2815  df-in 3217  df-ss 3224

This theorem is referenced by:  vdif0im  3574