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Theorem elintg 3774
 Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
Assertion
Ref Expression
elintg
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem elintg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eleq1 2200 . 2
2 eleq1 2200 . . 3
32ralbidv 2435 . 2
4 vex 2684 . . 3
54elint2 3773 . 2
61, 3, 5vtoclbg 2742 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wceq 1331   wcel 1480  wral 2414  cint 3766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-int 3767 This theorem is referenced by:  elinti  3775  elrint  3806  peano2  4504  pitonn  7649  peano1nnnn  7653  peano2nnnn  7654  1nn  8724  peano2nn  8725
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