Theorem pssv 4429

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 3988	. 2 ⊢ 𝐴 ⊆ V
2		dfpss2 4068	. 2 ⊢ (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V))
3	1, 2	mpbiran 709	1 ⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1540  Vcvv 3464   ⊆ wss 3931   ⊊ wpss 3932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-v 3466  df-ss 3948  df-pss 3951

This theorem is referenced by: (None)