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Theorem intnex 4386
 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 4385 . . . 4
21necon1bbii 2662 . . 3
3 inteq 4077 . . . 4
4 int0 4088 . . . 4
53, 4syl6eq 2490 . . 3
62, 5sylbi 189 . 2
7 vprc 4370 . . 3
8 eleq1 2502 . . 3
97, 8mtbiri 296 . 2
106, 9impbii 182 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 178   wceq 1653   wcel 1727  cvv 2962  c0 3613  cint 4074 This theorem is referenced by:  intabs  4390 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-v 2964  df-dif 3309  df-in 3313  df-ss 3320  df-nul 3614  df-int 4075
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