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

Theorem intnex 4786
 Description: If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
Assertion
Ref Expression
intnex 𝐴 ∈ V ↔ 𝐴 = V)

Proof of Theorem intnex
StepHypRef Expression
1 intex 4785 . . . 4 (𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
21necon1bbii 2839 . . 3 𝐴 ∈ V ↔ 𝐴 = ∅)
3 inteq 4448 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
4 int0 4460 . . . 4 ∅ = V
53, 4syl6eq 2671 . . 3 (𝐴 = ∅ → 𝐴 = V)
62, 5sylbi 207 . 2 𝐴 ∈ V → 𝐴 = V)
7 vprc 4761 . . 3 ¬ V ∈ V
8 eleq1 2686 . . 3 ( 𝐴 = V → ( 𝐴 ∈ V ↔ V ∈ V))
97, 8mtbiri 317 . 2 ( 𝐴 = V → ¬ 𝐴 ∈ V)
106, 9impbii 199 1 𝐴 ∈ V ↔ 𝐴 = V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1480   ∈ wcel 1987  Vcvv 3189  ∅c0 3896  ∩ cint 4445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3191  df-dif 3562  df-in 3566  df-ss 3573  df-nul 3897  df-int 4446 This theorem is referenced by:  intabs  4790  relintabex  37403
 Copyright terms: Public domain W3C validator