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Theorem limomss 4677
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 4467 . 2  |-  ( Lim 
A  ->  Ord  A )
2 ordeleqon 4596 . . 3  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
3 elom 4675 . . . . . . . . . 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 4420 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( Lim  y  <->  Lim  A ) )
6 eleq2 2357 . . . . . . . . . . 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 2881 . . . . . . . . 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 3198 . . . . 5  |-  ( ( A  e.  On  /\  Lim  A )  ->  om  C_  A
)
1312ex 423 . . . 4  |-  ( A  e.  On  ->  ( Lim  A  ->  om  C_  A
) )
14 omsson 4676 . . . . . 6  |-  om  C_  On
15 sseq2 3213 . . . . . 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 1530    = wceq 1632    e. wcel 1696    C_ wss 3165   Ord word 4407   Oncon0 4408   Lim wlim 4409   omcom 4672
This theorem is referenced by:  limom  4687  rdg0  6450  frfnom  6463  frsuc  6465  r1fin  7461  rankdmr1  7489  rankeq0b  7548  cardlim  7621  ackbij2  7885  cfom  7906  wunom  8358  inar1  8413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673
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