Theorem vss 3494

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 3201	. . 3 |- A C_ _V
2	1	biantrur 303	. 2 |- ( _V C_ A <-> ( A C_ _V /\ _V C_ A ) )
3		eqss 3194	. 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 1364   _Vcvv 2760   C_ wss 3153

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762  df-in 3159  df-ss 3166

This theorem is referenced by:  vdif0im  3512