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Theorem elom3 8171
Description: A simplification of elom 6714 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
Assertion
Ref Expression
elom3  |-  ( A  e.  om  <->  A. x
( Lim  x  ->  A  e.  x ) )
Distinct variable group:    x, A

Proof of Theorem elom3
StepHypRef Expression
1 elom 6714 . 2  |-  ( A  e.  om  <->  ( A  e.  On  /\  A. x
( Lim  x  ->  A  e.  x ) ) )
2 limom 6726 . . . . 5  |-  Lim  om
3 omex 8166 . . . . . 6  |-  om  e.  _V
4 limeq 5442 . . . . . . 7  |-  ( x  =  om  ->  ( Lim  x  <->  Lim  om ) )
5 eleq2 2538 . . . . . . 7  |-  ( x  =  om  ->  ( A  e.  x  <->  A  e.  om ) )
64, 5imbi12d 327 . . . . . 6  |-  ( x  =  om  ->  (
( Lim  x  ->  A  e.  x )  <->  ( Lim  om 
->  A  e.  om ) ) )
73, 6spcv 3126 . . . . 5  |-  ( A. x ( Lim  x  ->  A  e.  x )  ->  ( Lim  om  ->  A  e.  om )
)
82, 7mpi 20 . . . 4  |-  ( A. x ( Lim  x  ->  A  e.  x )  ->  A  e.  om )
9 nnon 6717 . . . 4  |-  ( A  e.  om  ->  A  e.  On )
108, 9syl 17 . . 3  |-  ( A. x ( Lim  x  ->  A  e.  x )  ->  A  e.  On )
1110pm4.71ri 645 . 2  |-  ( A. x ( Lim  x  ->  A  e.  x )  <-> 
( A  e.  On  /\ 
A. x ( Lim  x  ->  A  e.  x ) ) )
121, 11bitr4i 260 1  |-  ( A  e.  om  <->  A. x
( Lim  x  ->  A  e.  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    = wceq 1452    e. wcel 1904   Oncon0 5430   Lim wlim 5431   omcom 6711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-om 6712
This theorem is referenced by:  dfom4  8172  dfom5  8173
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