Theorem pssv 4368

Description: Any non-universal class is a proper subclass of the universal class. Dual of 0pss 4366. (Contributed by NM, 17-May-1998.)

Assertion

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

Proof of Theorem pssv

Step	Hyp	Ref	Expression
1		ssv 3960	. 2 ⊢ 𝐴 ⊆ V
2		dfpss2 4041	. 2 ⊢ (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V))
3	1, 2	mpbiran 721	1 ⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1569  Vcvv 3454   ⊆ wss 3904   ⊊ wpss 3905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-ss 3921  df-pss 3924

This theorem is used by: (None)