Theorem vss 3471

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 3178	. . 3 |- A C_ _V
2	1	biantrur 303	. 2 |- ( _V C_ A <-> ( A C_ _V /\ _V C_ A ) )
3		eqss 3171	. 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 1353   _Vcvv 2738   C_ wss 3130

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2740  df-in 3136  df-ss 3143

This theorem is referenced by:  vdif0im  3489