New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  elex22 GIF version

Theorem elex22 2870
 Description: If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
elex22 ((A B A C) → x(x B x C))
Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem elex22
StepHypRef Expression
1 eleq1a 2422 . . . 4 (A B → (x = Ax B))
2 eleq1a 2422 . . . 4 (A C → (x = Ax C))
31, 2anim12ii 553 . . 3 ((A B A C) → (x = A → (x B x C)))
43alrimiv 1631 . 2 ((A B A C) → x(x = A → (x B x C)))
5 elisset 2869 . . 3 (A Bx x = A)
65adantr 451 . 2 ((A B A C) → x x = A)
7 exim 1575 . 2 (x(x = A → (x B x C)) → (x x = Ax(x B x C)))
84, 6, 7sylc 56 1 ((A B A C) → x(x B x C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710 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-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator