New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  elelpwi GIF version

Theorem elelpwi 3732
 Description: If A belongs to a part of C then A belongs to C. (Contributed by FL, 3-Aug-2009.)
Assertion
Ref Expression
elelpwi ((A B B C) → A C)

Proof of Theorem elelpwi
StepHypRef Expression
1 elpwi 3730 . . 3 (B CB C)
21sseld 3272 . 2 (B C → (A BA C))
32impcom 419 1 ((A B B C) → A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ℘cpw 3722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pw 3724 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator