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Theorem elinel1 3266
 Description: Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elinel1 (𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)

Proof of Theorem elinel1
StepHypRef Expression
1 elin 3263 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21simplbi 272 1 (𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1481   ∩ cin 3074 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3081 This theorem is referenced by:  elin1d  3269  fival  6865  fi0  6870  blbas  12639  blres  12640  pilem3  12910  taupi  13428
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