Theorem pssv 3261

Description: Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)

Assertion

Ref	Expression
pssv	⊢ (A ⊊ V ↔ ¬ A = V)

Proof of Theorem pssv

Step	Hyp	Ref	Expression
1		ssv 2959	. 2 ⊢ A ⊆ V
2		dfpss2 3023	. 2 ⊢ (A ⊊ V ↔ (A ⊆ V ∧ ¬ A = V))
3	1, 2	mpbiran 846	1 ⊢ (A ⊊ V ↔ ¬ A = V)

Syntax hints:  ¬ wn 3   ↔ wb 98   = wceq 1242  Vcvv 2551   ⊆ wss 2911   ⊊ wpss 2912

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ne 2203  df-v 2553  df-in 2918  df-ss 2925  df-pss 2927

This theorem is referenced by: (None)