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Theorem elinel2 3783
Description: Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elinel2 (𝐴 ∈ (𝐵𝐶) → 𝐴𝐶)

Proof of Theorem elinel2
StepHypRef Expression
1 elin 3779 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21simprbi 480 1 (𝐴 ∈ (𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  cin 3558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-in 3566
This theorem is referenced by:  elin2d  3786  eldmeldmressn  5404  onfr  5727  ncvsge0  22872  itg2cnlem2  23448  uhgrspansubgrlem  26088  disjin2  29263  partfun  29336  xrge0tsmsd  29588  heicant  33103  mndoisexid  33327  fiinfi  37386  restuni3  38814  disjinfi  38877  inmap  38898  iocopn  39180  icoopn  39185  icomnfinre  39213  uzinico  39221  islpcn  39298  lptre2pt  39299  limcresiooub  39301  limcresioolb  39302  limclner  39310  limsupmnflem  39379  icccncfext  39426  stoweidlem39  39584  stoweidlem50  39595  stoweidlem57  39602  fourierdlem32  39684  fourierdlem33  39685  fourierdlem48  39699  fourierdlem49  39700  fourierdlem71  39722  fourierdlem80  39731  qndenserrnbllem  39842  sge0rnre  39909  sge0z  39920  sge0tsms  39925  sge0cl  39926  sge0f1o  39927  sge0fsum  39932  sge0sup  39936  sge0rnbnd  39938  sge0ltfirp  39945  sge0resplit  39951  sge0le  39952  sge0split  39954  sge0iunmptlemre  39960  sge0ltfirpmpt2  39971  sge0isum  39972  sge0xaddlem1  39978  sge0xaddlem2  39979  sge0pnffsumgt  39987  sge0gtfsumgt  39988  sge0uzfsumgt  39989  sge0seq  39991  sge0reuz  39992  meadjiunlem  40010  caragendifcl  40056  omeiunltfirp  40061  carageniuncllem2  40064  caratheodorylem2  40069  hspmbllem2  40169  pimiooltgt  40249  pimdecfgtioc  40253  pimincfltioc  40254  pimdecfgtioo  40255  pimincfltioo  40256  sssmf  40275  smfaddlem1  40299  smfaddlem2  40300  smfadd  40301  mbfpsssmf  40319  smfmullem4  40329  smfmul  40330  smfdiv  40332  smfsuplem1  40345  fmtno4prm  40807  rnghmsubcsetclem2  41285  rhmsubcsetclem2  41331  rhmsubcrngclem2  41337
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