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Theorem elom 6676
Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 8056. (Contributed by NM, 15-May-1994.)
Assertion
Ref Expression
elom  |-  ( A  e.  om  <->  ( A  e.  On  /\  A. x
( Lim  x  ->  A  e.  x ) ) )
Distinct variable group:    x, A

Proof of Theorem elom
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq1 2526 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
21imbi2d 314 . . 3  |-  ( y  =  A  ->  (
( Lim  x  ->  y  e.  x )  <->  ( Lim  x  ->  A  e.  x
) ) )
32albidv 1718 . 2  |-  ( y  =  A  ->  ( A. x ( Lim  x  ->  y  e.  x )  <->  A. x ( Lim  x  ->  A  e.  x ) ) )
4 df-om 6674 . 2  |-  om  =  { y  e.  On  |  A. x ( Lim  x  ->  y  e.  x ) }
53, 4elrab2 3256 1  |-  ( A  e.  om  <->  ( A  e.  On  /\  A. x
( Lim  x  ->  A  e.  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398    e. wcel 1823   Oncon0 4867   Lim wlim 4868   omcom 6673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-om 6674
This theorem is referenced by:  limomss  6678  ordom  6682  nnlim  6686  limom  6688  elom3  8056  dfom5b  29790
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