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Theorem eliin 3653
Description: Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
Assertion
Ref Expression
eliin  V  |^|_  C  C
Distinct variable group:   ,
Allowed substitution hints:   ()    C()    V()

Proof of Theorem eliin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . . 3  C  C
21ralbidv 2320 . 2  C  C
3 df-iin 3651 . 2  |^|_  C  {  |  C }
42, 3elab2g 2683 1  V  |^|_  C  C
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1242   wcel 1390  wral 2300   |^|_ciin 3649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-iin 3651
This theorem is referenced by:  iinconstm  3657  iuniin  3658  iinss1  3660  ssiinf  3697  iinss  3699  iinss2  3700  iinab  3709  iundif2ss  3713  iindif2m  3715  iinin2m  3716  elriin  3718  iinpw  3733  xpiindim  4416  cnviinm  4802  iinerm  6114
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