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Theorem elom 4659
Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 7349. (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 2343 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
21imbi2d 307 . . 3  |-  ( y  =  A  ->  (
( Lim  x  ->  y  e.  x )  <->  ( Lim  x  ->  A  e.  x
) ) )
32albidv 1611 . 2  |-  ( y  =  A  ->  ( A. x ( Lim  x  ->  y  e.  x )  <->  A. x ( Lim  x  ->  A  e.  x ) ) )
4 df-om 4657 . 2  |-  om  =  { y  e.  On  |  A. x ( Lim  x  ->  y  e.  x ) }
53, 4elrab2 2925 1  |-  ( A  e.  om  <->  ( A  e.  On  /\  A. x
( Lim  x  ->  A  e.  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   Oncon0 4392   Lim wlim 4393   omcom 4656
This theorem is referenced by:  limomss  4661  ordom  4665  nnlim  4669  limom  4671  elom3  7349  dfom5b  24452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-om 4657
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