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Theorem limomss 4552
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 4344 . 2  |-  ( Lim 
A  ->  Ord  A )
2 ordeleqon 4471 . . 3  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
3 elom 4550 . . . . . . . . . 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 4297 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( Lim  y  <->  Lim  A ) )
6 eleq2 2314 . . . . . . . . . . 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 2805 . . . . . . . . 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 3106 . . . . 5  |-  ( ( A  e.  On  /\  Lim  A )  ->  om  C_  A
)
1312ex 425 . . . 4  |-  ( A  e.  On  ->  ( Lim  A  ->  om  C_  A
) )
14 omsson 4551 . . . . . 6  |-  om  C_  On
15 sseq2 3121 . . . . . 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 3078   Ord word 4284   Oncon0 4285   Lim wlim 4286   omcom 4547
This theorem is referenced by:  limom  4562  rdg0  6320  frfnom  6333  frsuc  6335  r1fin  7329  rankdmr1  7357  rankeq0b  7416  cardlim  7489  ackbij2  7753  cfom  7774  wunom  8222  inar1  8277
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548
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