Theorem pssv 4365

Description: Any non-universal class is a proper subclass of the universal class. Dual of 0pss 4363. (Contributed by NM, 17-May-1998.)

Assertion

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

Proof of Theorem pssv

Step	Hyp	Ref	Expression
1		ssv 3958	. 2 ⊢ 𝐴 ⊆ V
2		dfpss2 4039	. 2 ⊢ (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V))
3	1, 2	mpbiran 722	1 ⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570  Vcvv 3453   ⊆ wss 3902   ⊊ wpss 3903

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3455  df-ss 3919  df-pss 3922

This theorem is used by: (None)