Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  elinel2 GIF version

Theorem elinel2 3290
 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 3286 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
21simprbi 273 1 (𝐴 ∈ (𝐵𝐶) → 𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 2125   ∩ cin 3097 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-in 3104 This theorem is referenced by:  elin2d  3293  fival  6903  blres  12781  limcresi  12982  pilem3  13051  taupi  13590
 Copyright terms: Public domain W3C validator