Theorem pssv 4361

Description: Any non-universal class is a proper subclass of the universal class. Dual of 0pss 4359. (Contributed by NM, 17-May-1998.)

Assertion

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

Proof of Theorem pssv

Step	Hyp	Ref	Expression
1		ssv 3932	. 2 ⊢ 𝐴 ⊆ V
2		dfpss2 4015	. 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 1569  Vcvv 3450   ⊆ wss 3895   ⊊ wpss 4013

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-ss 3896  df-pss 4014

This theorem is used by: (None)