Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  intv Structured version   Visualization version   GIF version

Theorem intv 4871
 Description: The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
Assertion
Ref Expression
intv V = ∅

Proof of Theorem intv
StepHypRef Expression
1 0ex 4823 . 2 ∅ ∈ V
2 int0el 4540 . 2 (∅ ∈ V → V = ∅)
31, 2ax-mp 5 1 V = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ∅c0 3948  ∩ cint 4507 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-int 4508 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator