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

Theorem elunii 3896
 Description: Membership in class union. (Contributed by NM, 24-Mar-1995.)
Assertion
Ref Expression
elunii ((A B B C) → A C)

Proof of Theorem elunii
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2414 . . . . 5 (x = B → (A xA B))
2 eleq1 2413 . . . . 5 (x = B → (x CB C))
31, 2anbi12d 691 . . . 4 (x = B → ((A x x C) ↔ (A B B C)))
43spcegv 2940 . . 3 (B C → ((A B B C) → x(A x x C)))
54anabsi7 792 . 2 ((A B B C) → x(A x x C))
6 eluni 3894 . 2 (A Cx(A x x C))
75, 6sylibr 203 1 ((A B B C) → A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∪cuni 3891 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-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-uni 3892 This theorem is referenced by:  ssuni  3913  unipw  4117  nnadjoin  4520  sfinltfin  4535  nulnnn  4556
 Copyright terms: Public domain W3C validator