Theorem pssv 1732

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 1557	. 2 |- (A (. V <-> (A (_ V /\ -. A = V))
2		ssv 1520	. 2 |- A (_ V
3	1, 2	mpbiran 547	1 |- (A (. V <-> -. A = V)

Syntax hints:  -. wn 1   <-> wb 127   = wceq 1091  Vcvv 1348   (_ wss 1487   (. wpss 1488

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-v 1349  df-in 1491  df-ss 1492  df-pss 1494

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