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

Theorem intnex 4558
 Description: If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
Assertion
Ref Expression
intnex

Proof of Theorem intnex
StepHypRef Expression
1 intex 4557 . . . 4
21necon1bbii 2692 . . 3
3 inteq 4229 . . . 4
4 int0 4240 . . . 4
53, 4syl6eq 2521 . . 3
62, 5sylbi 200 . 2
7 vprc 4534 . . 3
8 eleq1 2537 . . 3
97, 8mtbiri 310 . 2
106, 9impbii 192 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 189   wceq 1452   wcel 1904  cvv 3031  c0 3722  cint 4226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-int 4227 This theorem is referenced by:  intabs  4562  relintabex  36258
 Copyright terms: Public domain W3C validator