Theorem pssv 2310

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 2133	. 2 |- (A (. V <-> (A (_ V /\ -. A = V))
2		ssv 2081	. 2 |- A (_ V
3	1, 2	mpbiran 728	1 |- (A (. V <-> -. A = V)

Syntax hints:  -. wn 2   <-> wb 146   = wceq 956  Vcvv 1811   (_ wss 2047   (. wpss 2048

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-in 2051  df-ss 2053  df-pss 2055

Copyright terms: Public domain