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Theorem elini 4167
Description: Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
elini.1 𝐴𝐵
elini.2 𝐴𝐶
Assertion
Ref Expression
elini 𝐴 ∈ (𝐵𝐶)

Proof of Theorem elini
StepHypRef Expression
1 elini.1 . 2 𝐴𝐵
2 elini.2 . 2 𝐴𝐶
3 elin 4166 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
41, 2, 3mpbir2an 707 1 𝐴 ∈ (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  cin 3932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940
This theorem is referenced by:  isfin1-3  9796  isdrs2  17537  fpwipodrs  17762  0cmp  21930  comppfsc  22068  ptcmpfi  22349  alexsubALTlem2  22584  alexsubALTlem4  22586  ptcmp  22594  cnstrcvs  23672  cncvs  23676  recvs  23677  qcvs  23678  cnncvs  23690  ovolicc1  24044  ioorf  24101  corclrcl  39930  0pwfi  41198  sge0rnn0  42527  sge0reuz  42606
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