Theorem vss 4397

Description: Only the universal class has the universal class as a subclass. (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 3958	. . 3 ⊢ 𝐴 ⊆ V
2	1	biantrur 538	. 2 ⊢ (V ⊆ 𝐴 ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
3		eqss 3949	. 2 ⊢ (𝐴 = V ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴))
4	2, 3	bitr4i 280	1 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559  Vcvv 3453   ⊆ wss 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3919

This theorem is referenced by:  vdif0  4420  fineqvr1ombregs  35394