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Theorem elini 3170
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 3169 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
41, 2, 3mpbir2an 886 1 𝐴 ∈ (𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wcel 1436  cin 2985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-in 2992
This theorem is referenced by: (None)
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