Theorem pssv 4448

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 4007	. 2 ⊢ 𝐴 ⊆ V
2		dfpss2 4087	. 2 ⊢ (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V))
3	1, 2	mpbiran 709	1 ⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1539  Vcvv 3479   ⊆ wss 3950   ⊊ wpss 3951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-v 3481  df-ss 3967  df-pss 3970

This theorem is referenced by: (None)