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Theorem limomss 4663
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 4453 . 2  |-  ( Lim 
A  ->  Ord  A )
2 ordeleqon 4582 . . 3  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
3 elom 4661 . . . . . . . . . 10  |-  ( x  e.  om  <->  ( x  e.  On  /\  A. y
( Lim  y  ->  x  e.  y ) ) )
43simprbi 450 . . . . . . . . 9  |-  ( x  e.  om  ->  A. y
( Lim  y  ->  x  e.  y ) )
5 limeq 4406 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( Lim  y  <->  Lim  A ) )
6 eleq2 2346 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
x  e.  y  <->  x  e.  A ) )
75, 6imbi12d 311 . . . . . . . . . 10  |-  ( y  =  A  ->  (
( Lim  y  ->  x  e.  y )  <->  ( Lim  A  ->  x  e.  A
) ) )
87spcgv 2870 . . . . . . . . 9  |-  ( A  e.  On  ->  ( A. y ( Lim  y  ->  x  e.  y )  ->  ( Lim  A  ->  x  e.  A ) ) )
94, 8syl5 28 . . . . . . . 8  |-  ( A  e.  On  ->  (
x  e.  om  ->  ( Lim  A  ->  x  e.  A ) ) )
109com23 72 . . . . . . 7  |-  ( A  e.  On  ->  ( Lim  A  ->  ( x  e.  om  ->  x  e.  A ) ) )
1110imp 418 . . . . . 6  |-  ( ( A  e.  On  /\  Lim  A )  ->  (
x  e.  om  ->  x  e.  A ) )
1211ssrdv 3187 . . . . 5  |-  ( ( A  e.  On  /\  Lim  A )  ->  om  C_  A
)
1312ex 423 . . . 4  |-  ( A  e.  On  ->  ( Lim  A  ->  om  C_  A
) )
14 omsson 4662 . . . . . 6  |-  om  C_  On
15 sseq2 3202 . . . . . 6  |-  ( A  =  On  ->  ( om  C_  A  <->  om  C_  On ) )
1614, 15mpbiri 224 . . . . 5  |-  ( A  =  On  ->  om  C_  A
)
1716a1d 22 . . . 4  |-  ( A  =  On  ->  ( Lim  A  ->  om  C_  A
) )
1813, 17jaoi 368 . . 3  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( Lim  A  ->  om  C_  A ) )
192, 18sylbi 187 . 2  |-  ( Ord 
A  ->  ( Lim  A  ->  om  C_  A ) )
201, 19mpcom 32 1  |-  ( Lim 
A  ->  om  C_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   A.wal 1529    = wceq 1625    e. wcel 1686    C_ wss 3154   Ord word 4393   Oncon0 4394   Lim wlim 4395   omcom 4658
This theorem is referenced by:  limom  4673  rdg0  6436  frfnom  6449  frsuc  6451  r1fin  7447  rankdmr1  7475  rankeq0b  7534  cardlim  7607  ackbij2  7871  cfom  7892  wunom  8344  inar1  8399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4307  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659
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