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Theorem elini 3759
 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 . . 3 𝐴𝐵
2 elini.2 . . 3 𝐴𝐶
31, 2pm3.2i 470 . 2 (𝐴𝐵𝐴𝐶)
4 elin 3758 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
53, 4mpbir 220 1 𝐴 ∈ (𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   ∈ wcel 1977   ∩ cin 3539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547 This theorem is referenced by:  recvs  22754  qcvs  22755  cnncvs  22767  0pwfi  38252  sge0rnn0  39261  sge0reuz  39340
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