Theorem pssv 3267

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 2965	. 2 ⊢ 𝐴 ⊆ V
2		dfpss2 3029	. 2 ⊢ (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V))
3	1, 2	mpbiran 847	1 ⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V)

Syntax hints:  ¬ wn 3   ↔ wb 98   = wceq 1243  Vcvv 2557   ⊆ wss 2917   ⊊ wpss 2918

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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-v 2559  df-in 2924  df-ss 2931  df-pss 2933

This theorem is referenced by: (None)