MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elom Structured version   Unicode version

Theorem elom 4850
Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 7605. (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 2498 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
21imbi2d 309 . . 3  |-  ( y  =  A  ->  (
( Lim  x  ->  y  e.  x )  <->  ( Lim  x  ->  A  e.  x
) ) )
32albidv 1636 . 2  |-  ( y  =  A  ->  ( A. x ( Lim  x  ->  y  e.  x )  <->  A. x ( Lim  x  ->  A  e.  x ) ) )
4 df-om 4848 . 2  |-  om  =  { y  e.  On  |  A. x ( Lim  x  ->  y  e.  x ) }
53, 4elrab2 3096 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 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   Oncon0 4583   Lim wlim 4584   omcom 4847
This theorem is referenced by:  limomss  4852  ordom  4856  nnlim  4860  limom  4862  elom3  7605  dfom5b  25759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-om 4848
  Copyright terms: Public domain W3C validator