Theorem vss 4364

Description: Only the universal class has the universal class as a subclass. Dual of ss0b 4357. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
vss	⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)

Proof of Theorem vss

Step	Hyp	Ref	Expression
1		ssv 3960	. . 3 ⊢ 𝐴 ⊆ V
2	1	biantrur 539	. 2 ⊢ (V ⊆ 𝐴 ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
3		eqss 3951	. 2 ⊢ (𝐴 = V ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
4	2, 3	bitr4i 281	1 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569  Vcvv 3454   ⊆ wss 3904

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-v 3456  df-ss 3921

This theorem is used by:  vvin  4365  vdif0  4428  fineqvr1ombregs  35559