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Theorem limomss 4598
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
StepHypRef Expression
1 limord 4388 . 2  |-  ( Lim 
A  ->  Ord  A )
2 ordeleqon 4517 . . 3  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
3 elom 4596 . . . . . . . . . 10  |-  ( x  e.  om  <->  ( x  e.  On  /\  A. y
( Lim  y  ->  x  e.  y ) ) )
43simprbi 452 . . . . . . . . 9  |-  ( x  e.  om  ->  A. y
( Lim  y  ->  x  e.  y ) )
5 limeq 4341 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( Lim  y  <->  Lim  A ) )
6 eleq2 2317 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
x  e.  y  <->  x  e.  A ) )
75, 6imbi12d 313 . . . . . . . . . 10  |-  ( y  =  A  ->  (
( Lim  y  ->  x  e.  y )  <->  ( Lim  A  ->  x  e.  A
) ) )
87cla4gv 2819 . . . . . . . . 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 420 . . . . . 6  |-  ( ( A  e.  On  /\  Lim  A )  ->  (
x  e.  om  ->  x  e.  A ) )
1211ssrdv 3127 . . . . 5  |-  ( ( A  e.  On  /\  Lim  A )  ->  om  C_  A
)
1312ex 425 . . . 4  |-  ( A  e.  On  ->  ( Lim  A  ->  om  C_  A
) )
14 omsson 4597 . . . . . 6  |-  om  C_  On
15 sseq2 3142 . . . . . 6  |-  ( A  =  On  ->  ( om  C_  A  <->  om  C_  On ) )
1614, 15mpbiri 226 . . . . 5  |-  ( A  =  On  ->  om  C_  A
)
1716a1d 24 . . . 4  |-  ( A  =  On  ->  ( Lim  A  ->  om  C_  A
) )
1813, 17jaoi 370 . . 3  |-  ( ( A  e.  On  \/  A  =  On )  ->  ( Lim  A  ->  om  C_  A ) )
192, 18sylbi 189 . 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 6    \/ wo 359    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621    C_ wss 3094   Ord word 4328   Oncon0 4329   Lim wlim 4330   omcom 4593
This theorem is referenced by:  limom  4608  rdg0  6367  frfnom  6380  frsuc  6382  r1fin  7378  rankdmr1  7406  rankeq0b  7465  cardlim  7538  ackbij2  7802  cfom  7823  wunom  8275  inar1  8330
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-tr 4054  df-eprel 4242  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594
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