Theorem vss 2908

Description: Only the universal class has the universal class as a subclass. (The proof was 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 2636	. . 3 |- A C_ _V
2	1	biantrur 794	. 2 |- (_V C_ A <-> (A C_ _V /\ _V C_ A))
3		eqss 2631	. 2 |- (A = _V <-> (A C_ _V /\ _V C_ A))
4	2, 3	bitr4i 193	1 |- (_V C_ A <-> A = _V)

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298  _Vcvv 2292   C_ wss 2593

This theorem is referenced by:  vdif0 2935  dmen 5466

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-ss 2605

Copyright terms: Public domain