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Theorem limomss 4783
Description: The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.)
Assertion
Ref Expression
limomss  |-  ( Lim 
A  ->  om  C_  A
)

Proof of Theorem limomss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limord 4574 . 2  |-  ( Lim 
A  ->  Ord  A )
2 ordeleqon 4702 . . 3  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
3 elom 4781 . . . . . . . . . 10  |-  ( x  e.  om  <->  ( x  e.  On  /\  A. y
( Lim  y  ->  x  e.  y ) ) )
43simprbi 451 . . . . . . . . 9  |-  ( x  e.  om  ->  A. y
( Lim  y  ->  x  e.  y ) )
5 limeq 4527 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( Lim  y  <->  Lim  A ) )
6 eleq2 2441 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
x  e.  y  <->  x  e.  A ) )
75, 6imbi12d 312 . . . . . . . . . 10  |-  ( y  =  A  ->  (
( Lim  y  ->  x  e.  y )  <->  ( Lim  A  ->  x  e.  A
) ) )
87spcgv 2972 . . . . . . . . 9  |-  ( A  e.  On  ->  ( A. y ( Lim  y  ->  x  e.  y )  ->  ( Lim  A  ->  x  e.  A ) ) )
94, 8syl5 30 . . . . . . . 8  |-  ( A  e.  On  ->  (
x  e.  om  ->  ( Lim  A  ->  x  e.  A ) ) )
109com23 74 . . . . . . 7  |-  ( A  e.  On  ->  ( Lim  A  ->  ( x  e.  om  ->  x  e.  A ) ) )
1110imp 419 . . . . . 6  |-  ( ( A  e.  On  /\  Lim  A )  ->  (
x  e.  om  ->  x  e.  A ) )
1211ssrdv 3290 . . . . 5  |-  ( ( A  e.  On  /\  Lim  A )  ->  om  C_  A
)
1312ex 424 . . . 4  |-  ( A  e.  On  ->  ( Lim  A  ->  om  C_  A
) )
14 omsson 4782 . . . . . 6  |-  om  C_  On
15 sseq2 3306 . . . . . 6  |-  ( A  =  On  ->  ( om  C_  A  <->  om  C_  On ) )
1614, 15mpbiri 225 . . . . 5  |-  ( A  =  On  ->  om  C_  A
)
1716a1d 23 . . . 4  |-  ( A  =  On  ->  ( Lim  A  ->  om  C_  A
) )
1813, 17jaoi 369 . . 3  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( Lim  A  ->  om  C_  A ) )
192, 18sylbi 188 . 2  |-  ( Ord 
A  ->  ( Lim  A  ->  om  C_  A ) )
201, 19mpcom 34 1  |-  ( Lim 
A  ->  om  C_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717    C_ wss 3256   Ord word 4514   Oncon0 4515   Lim wlim 4516   omcom 4778
This theorem is referenced by:  limom  4793  rdg0  6608  frfnom  6621  frsuc  6623  r1fin  7625  rankdmr1  7653  rankeq0b  7712  cardlim  7785  ackbij2  8049  cfom  8070  wunom  8521  inar1  8576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779
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