Theorem vss 4012

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 3625	. . 3 |- A C_ _V
2	1	biantrur 527	. 2 |- ( _V C_ A <-> ( A C_ _V /\ _V C_ A ) )
3		eqss 3618	. 2 |- ( A = _V <-> ( A C_ _V /\ _V C_ A ) )
4	2, 3	bitr4i 267	1 |- ( _V C_ A <-> A = _V )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   _Vcvv 3200   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by:  vdif0  4037