Theorem pssv 4399

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 3956	. 2 ⊢ 𝐴 ⊆ V
2		dfpss2 4038	. 2 ⊢ (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V))
3	1, 2	mpbiran 709	1 ⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1541  Vcvv 3438   ⊆ wss 3899   ⊊ wpss 3900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-v 3440  df-ss 3916  df-pss 3919

This theorem is referenced by: (None)