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Theorem cardlim 7605
Description: An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)
Assertion
Ref Expression
cardlim  |-  ( om  C_  ( card `  A
)  <->  Lim  ( card `  A
) )

Proof of Theorem cardlim
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sseq2 3200 . . . . . . . . . . 11  |-  ( (
card `  A )  =  suc  x  ->  ( om  C_  ( card `  A
)  <->  om  C_  suc  x ) )
21biimpd 198 . . . . . . . . . 10  |-  ( (
card `  A )  =  suc  x  ->  ( om  C_  ( card `  A
)  ->  om  C_  suc  x ) )
3 limom 4671 . . . . . . . . . . . 12  |-  Lim  om
4 limsssuc 4641 . . . . . . . . . . . 12  |-  ( Lim 
om  ->  ( om  C_  x  <->  om  C_  suc  x ) )
53, 4ax-mp 8 . . . . . . . . . . 11  |-  ( om  C_  x  <->  om  C_  suc  x )
6 infensuc 7039 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  om  C_  x )  ->  x  ~~  suc  x )
76ex 423 . . . . . . . . . . 11  |-  ( x  e.  On  ->  ( om  C_  x  ->  x  ~~  suc  x ) )
85, 7syl5bir 209 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( om  C_  suc  x  ->  x  ~~  suc  x ) )
92, 8sylan9r 639 . . . . . . . . 9  |-  ( ( x  e.  On  /\  ( card `  A )  =  suc  x )  -> 
( om  C_  ( card `  A )  ->  x  ~~  suc  x ) )
10 breq2 4027 . . . . . . . . . 10  |-  ( (
card `  A )  =  suc  x  ->  (
x  ~~  ( card `  A )  <->  x  ~~  suc  x ) )
1110adantl 452 . . . . . . . . 9  |-  ( ( x  e.  On  /\  ( card `  A )  =  suc  x )  -> 
( x  ~~  ( card `  A )  <->  x  ~~  suc  x ) )
129, 11sylibrd 225 . . . . . . . 8  |-  ( ( x  e.  On  /\  ( card `  A )  =  suc  x )  -> 
( om  C_  ( card `  A )  ->  x  ~~  ( card `  A
) ) )
1312ex 423 . . . . . . 7  |-  ( x  e.  On  ->  (
( card `  A )  =  suc  x  ->  ( om  C_  ( card `  A
)  ->  x  ~~  ( card `  A )
) ) )
1413com3r 73 . . . . . 6  |-  ( om  C_  ( card `  A
)  ->  ( x  e.  On  ->  ( ( card `  A )  =  suc  x  ->  x  ~~  ( card `  A
) ) ) )
1514imp 418 . . . . 5  |-  ( ( om  C_  ( card `  A )  /\  x  e.  On )  ->  (
( card `  A )  =  suc  x  ->  x  ~~  ( card `  A
) ) )
16 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
1716sucid 4471 . . . . . . . . 9  |-  x  e. 
suc  x
18 eleq2 2344 . . . . . . . . 9  |-  ( (
card `  A )  =  suc  x  ->  (
x  e.  ( card `  A )  <->  x  e.  suc  x ) )
1917, 18mpbiri 224 . . . . . . . 8  |-  ( (
card `  A )  =  suc  x  ->  x  e.  ( card `  A
) )
20 cardidm 7592 . . . . . . . 8  |-  ( card `  ( card `  A
) )  =  (
card `  A )
2119, 20syl6eleqr 2374 . . . . . . 7  |-  ( (
card `  A )  =  suc  x  ->  x  e.  ( card `  ( card `  A ) ) )
22 cardne 7598 . . . . . . 7  |-  ( x  e.  ( card `  ( card `  A ) )  ->  -.  x  ~~  ( card `  A )
)
2321, 22syl 15 . . . . . 6  |-  ( (
card `  A )  =  suc  x  ->  -.  x  ~~  ( card `  A
) )
2423a1i 10 . . . . 5  |-  ( ( om  C_  ( card `  A )  /\  x  e.  On )  ->  (
( card `  A )  =  suc  x  ->  -.  x  ~~  ( card `  A
) ) )
2515, 24pm2.65d 166 . . . 4  |-  ( ( om  C_  ( card `  A )  /\  x  e.  On )  ->  -.  ( card `  A )  =  suc  x )
2625nrexdv 2646 . . 3  |-  ( om  C_  ( card `  A
)  ->  -.  E. x  e.  On  ( card `  A
)  =  suc  x
)
27 peano1 4675 . . . . . 6  |-  (/)  e.  om
28 ssel 3174 . . . . . 6  |-  ( om  C_  ( card `  A
)  ->  ( (/)  e.  om  -> 
(/)  e.  ( card `  A ) ) )
2927, 28mpi 16 . . . . 5  |-  ( om  C_  ( card `  A
)  ->  (/)  e.  (
card `  A )
)
30 n0i 3460 . . . . 5  |-  ( (/)  e.  ( card `  A
)  ->  -.  ( card `  A )  =  (/) )
31 cardon 7577 . . . . . . . . 9  |-  ( card `  A )  e.  On
3231onordi 4497 . . . . . . . 8  |-  Ord  ( card `  A )
33 ordzsl 4636 . . . . . . . 8  |-  ( Ord  ( card `  A
)  <->  ( ( card `  A )  =  (/)  \/ 
E. x  e.  On  ( card `  A )  =  suc  x  \/  Lim  ( card `  A )
) )
3432, 33mpbi 199 . . . . . . 7  |-  ( (
card `  A )  =  (/)  \/  E. x  e.  On  ( card `  A
)  =  suc  x  \/  Lim  ( card `  A
) )
35 3orass 937 . . . . . . 7  |-  ( ( ( card `  A
)  =  (/)  \/  E. x  e.  On  ( card `  A )  =  suc  x  \/  Lim  ( card `  A )
)  <->  ( ( card `  A )  =  (/)  \/  ( E. x  e.  On  ( card `  A
)  =  suc  x  \/  Lim  ( card `  A
) ) ) )
3634, 35mpbi 199 . . . . . 6  |-  ( (
card `  A )  =  (/)  \/  ( E. x  e.  On  ( card `  A )  =  suc  x  \/  Lim  ( card `  A )
) )
3736ori 364 . . . . 5  |-  ( -.  ( card `  A
)  =  (/)  ->  ( E. x  e.  On  ( card `  A )  =  suc  x  \/  Lim  ( card `  A )
) )
3829, 30, 373syl 18 . . . 4  |-  ( om  C_  ( card `  A
)  ->  ( E. x  e.  On  ( card `  A )  =  suc  x  \/  Lim  ( card `  A )
) )
3938ord 366 . . 3  |-  ( om  C_  ( card `  A
)  ->  ( -.  E. x  e.  On  ( card `  A )  =  suc  x  ->  Lim  ( card `  A )
) )
4026, 39mpd 14 . 2  |-  ( om  C_  ( card `  A
)  ->  Lim  ( card `  A ) )
41 limomss 4661 . 2  |-  ( Lim  ( card `  A
)  ->  om  C_  ( card `  A ) )
4240, 41impbii 180 1  |-  ( om  C_  ( card `  A
)  <->  Lim  ( card `  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   (/)c0 3455   class class class wbr 4023   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394   omcom 4656   ` cfv 5255    ~~ cen 6860   cardccrd 7568
This theorem is referenced by:  infxpenlem  7641  alephislim  7710  cflim2  7889  winalim  8317  gruina  8440  cartarlim  25905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-card 7572
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