Theorem pssv 2314

Description: Any non-universal class is a proper subclass of the universal class.

Assertion

Ref	Expression
pssv	|- (A (. V <-> -. A = V)

Proof of Theorem pssv

Step	Hyp	Ref	Expression
1		dfpss2 2136	. 2 |- (A (. V <-> (A (_ V /\ -. A = V))
2		ssv 2084	. 2 |- A (_ V
3	1, 2	mpbiran 730	1 |- (A (. V <-> -. A = V)

Syntax hints:  -. wn 2   <-> wb 146   = wceq 958  Vcvv 1814   (_ wss 2050   (. wpss 2051

This theorem is referenced by:  vxveqv 10467

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-in 2054  df-ss 2056  df-pss 2058

Copyright terms: Public domain