Theorem vss 3588

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 3292	. . 3 ⊢ A ⊆ V
2	1	biantrur 492	. 2 ⊢ (V ⊆ A ↔ (A ⊆ V ∧ V ⊆ A))
3		eqss 3288	. 2 ⊢ (A = V ↔ (A ⊆ V ∧ V ⊆ A))
4	2, 3	bitr4i 243	1 ⊢ (V ⊆ A ↔ A = V)

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642  Vcvv 2860   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  vdif0  3611  dmen  6042