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Theorem inssdif0 3695
 Description: Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
Assertion
Ref Expression
inssdif0

Proof of Theorem inssdif0
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3530 . . . . . 6
21imbi1i 316 . . . . 5
3 iman 414 . . . . 5
42, 3bitri 241 . . . 4
5 eldif 3330 . . . . . 6
65anbi2i 676 . . . . 5
7 elin 3530 . . . . 5
8 anass 631 . . . . 5
96, 7, 83bitr4ri 270 . . . 4
104, 9xchbinx 302 . . 3
1110albii 1575 . 2
12 dfss2 3337 . 2
13 eq0 3642 . 2
1411, 12, 133bitr4i 269 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359  wal 1549   wceq 1652   wcel 1725   cdif 3317   cin 3319   wss 3320  c0 3628 This theorem is referenced by:  disjdif  3700  inf3lem3  7585  ssfin4  8190  isnrm2  17422  1stccnp  17525  llycmpkgen2  17582  ufileu  17951  fclscf  18057  flimfnfcls  18060  opnbnd  26328  diophrw  26817  setindtr  27095 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629
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