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

Theorem intnex 4851
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 4850 . . . 4 (𝐴 ≠ ∅ ↔ 𝐴 ∈ V)
21necon1bbii 2872 . . 3 𝐴 ∈ V ↔ 𝐴 = ∅)
3 inteq 4510 . . . 4 (𝐴 = ∅ → 𝐴 = ∅)
4 int0 4522 . . . 4 ∅ = V
53, 4syl6eq 2701 . . 3 (𝐴 = ∅ → 𝐴 = V)
62, 5sylbi 207 . 2 𝐴 ∈ V → 𝐴 = V)
7 vprc 4829 . . 3 ¬ V ∈ V
8 eleq1 2718 . . 3 ( 𝐴 = V → ( 𝐴 ∈ V ↔ V ∈ V))
97, 8mtbiri 316 . 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 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-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814
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-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-int 4508
This theorem is referenced by:  intabs  4855  relintabex  38204
  Copyright terms: Public domain W3C validator