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Theorem cardlim 7489
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
StepHypRef Expression
1 sseq2 3121 . . . . . . . . . . 11  |-  ( (
card `  A )  =  suc  x  ->  ( om  C_  ( card `  A
)  <->  om  C_  suc  x ) )
21biimpd 200 . . . . . . . . . 10  |-  ( (
card `  A )  =  suc  x  ->  ( om  C_  ( card `  A
)  ->  om  C_  suc  x ) )
3 limom 4562 . . . . . . . . . . . 12  |-  Lim  om
4 limsssuc 4532 . . . . . . . . . . . 12  |-  ( Lim 
om  ->  ( om  C_  x  <->  om  C_  suc  x ) )
53, 4ax-mp 10 . . . . . . . . . . 11  |-  ( om  C_  x  <->  om  C_  suc  x )
6 infensuc 6924 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  om  C_  x )  ->  x  ~~  suc  x )
76ex 425 . . . . . . . . . . 11  |-  ( x  e.  On  ->  ( om  C_  x  ->  x  ~~  suc  x ) )
85, 7syl5bir 211 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( om  C_  suc  x  ->  x  ~~  suc  x ) )
92, 8sylan9r 642 . . . . . . . . 9  |-  ( ( x  e.  On  /\  ( card `  A )  =  suc  x )  -> 
( om  C_  ( card `  A )  ->  x  ~~  suc  x ) )
10 breq2 3924 . . . . . . . . . 10  |-  ( (
card `  A )  =  suc  x  ->  (
x  ~~  ( card `  A )  <->  x  ~~  suc  x ) )
1110adantl 454 . . . . . . . . 9  |-  ( ( x  e.  On  /\  ( card `  A )  =  suc  x )  -> 
( x  ~~  ( card `  A )  <->  x  ~~  suc  x ) )
129, 11sylibrd 227 . . . . . . . 8  |-  ( ( x  e.  On  /\  ( card `  A )  =  suc  x )  -> 
( om  C_  ( card `  A )  ->  x  ~~  ( card `  A
) ) )
1312ex 425 . . . . . . 7  |-  ( x  e.  On  ->  (
( card `  A )  =  suc  x  ->  ( om  C_  ( card `  A
)  ->  x  ~~  ( card `  A )
) ) )
1413com3r 75 . . . . . 6  |-  ( om  C_  ( card `  A
)  ->  ( x  e.  On  ->  ( ( card `  A )  =  suc  x  ->  x  ~~  ( card `  A
) ) ) )
1514imp 420 . . . . 5  |-  ( ( om  C_  ( card `  A )  /\  x  e.  On )  ->  (
( card `  A )  =  suc  x  ->  x  ~~  ( card `  A
) ) )
16 vex 2730 . . . . . . . . . 10  |-  x  e. 
_V
1716sucid 4364 . . . . . . . . 9  |-  x  e. 
suc  x
18 eleq2 2314 . . . . . . . . 9  |-  ( (
card `  A )  =  suc  x  ->  (
x  e.  ( card `  A )  <->  x  e.  suc  x ) )
1917, 18mpbiri 226 . . . . . . . 8  |-  ( (
card `  A )  =  suc  x  ->  x  e.  ( card `  A
) )
20 cardidm 7476 . . . . . . . 8  |-  ( card `  ( card `  A
) )  =  (
card `  A )
2119, 20syl6eleqr 2344 . . . . . . 7  |-  ( (
card `  A )  =  suc  x  ->  x  e.  ( card `  ( card `  A ) ) )
22 cardne 7482 . . . . . . 7  |-  ( x  e.  ( card `  ( card `  A ) )  ->  -.  x  ~~  ( card `  A )
)
2321, 22syl 17 . . . . . 6  |-  ( (
card `  A )  =  suc  x  ->  -.  x  ~~  ( card `  A
) )
2423a1i 12 . . . . 5  |-  ( ( om  C_  ( card `  A )  /\  x  e.  On )  ->  (
( card `  A )  =  suc  x  ->  -.  x  ~~  ( card `  A
) ) )
2515, 24pm2.65d 168 . . . 4  |-  ( ( om  C_  ( card `  A )  /\  x  e.  On )  ->  -.  ( card `  A )  =  suc  x )
2625nrexdv 2608 . . 3  |-  ( om  C_  ( card `  A
)  ->  -.  E. x  e.  On  ( card `  A
)  =  suc  x
)
27 peano1 4566 . . . . . 6  |-  (/)  e.  om
28 ssel 3097 . . . . . 6  |-  ( om  C_  ( card `  A
)  ->  ( (/)  e.  om  -> 
(/)  e.  ( card `  A ) ) )
2927, 28mpi 18 . . . . 5  |-  ( om  C_  ( card `  A
)  ->  (/)  e.  (
card `  A )
)
30 n0i 3367 . . . . 5  |-  ( (/)  e.  ( card `  A
)  ->  -.  ( card `  A )  =  (/) )
31 cardon 7461 . . . . . . . . 9  |-  ( card `  A )  e.  On
3231onordi 4388 . . . . . . . 8  |-  Ord  ( card `  A )
33 ordzsl 4527 . . . . . . . 8  |-  ( Ord  ( card `  A
)  <->  ( ( card `  A )  =  (/)  \/ 
E. x  e.  On  ( card `  A )  =  suc  x  \/  Lim  ( card `  A )
) )
3432, 33mpbi 201 . . . . . . 7  |-  ( (
card `  A )  =  (/)  \/  E. x  e.  On  ( card `  A
)  =  suc  x  \/  Lim  ( card `  A
) )
35 3orass 942 . . . . . . 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 201 . . . . . 6  |-  ( (
card `  A )  =  (/)  \/  ( E. x  e.  On  ( card `  A )  =  suc  x  \/  Lim  ( card `  A )
) )
3736ori 366 . . . . 5  |-  ( -.  ( card `  A
)  =  (/)  ->  ( E. x  e.  On  ( card `  A )  =  suc  x  \/  Lim  ( card `  A )
) )
3829, 30, 373syl 20 . . . 4  |-  ( om  C_  ( card `  A
)  ->  ( E. x  e.  On  ( card `  A )  =  suc  x  \/  Lim  ( card `  A )
) )
3938ord 368 . . 3  |-  ( om  C_  ( card `  A
)  ->  ( -.  E. x  e.  On  ( card `  A )  =  suc  x  ->  Lim  ( card `  A )
) )
4026, 39mpd 16 . 2  |-  ( om  C_  ( card `  A
)  ->  Lim  ( card `  A ) )
41 limomss 4552 . 2  |-  ( Lim  ( card `  A
)  ->  om  C_  ( card `  A ) )
4240, 41impbii 182 1  |-  ( om  C_  ( card `  A
)  <->  Lim  ( card `  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 938    = wceq 1619    e. wcel 1621   E.wrex 2510    C_ wss 3078   (/)c0 3362   class class class wbr 3920   Ord word 4284   Oncon0 4285   Lim wlim 4286   suc csuc 4287   omcom 4547   ` cfv 4592    ~~ cen 6746   cardccrd 7452
This theorem is referenced by:  infxpenlem  7525  alephislim  7594  cflim2  7773  winalim  8197  gruina  8320  cartarlim  25071
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-pow 4082  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-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  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  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-1o 6365  df-er 6546  df-en 6750  df-dom 6751  df-card 7456
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