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Theorem cardlim 8138
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 3375 . . . . . . . . . . 11  |-  ( (
card `  A )  =  suc  x  ->  ( om  C_  ( card `  A
)  <->  om  C_  suc  x ) )
21biimpd 207 . . . . . . . . . 10  |-  ( (
card `  A )  =  suc  x  ->  ( om  C_  ( card `  A
)  ->  om  C_  suc  x ) )
3 limom 6490 . . . . . . . . . . . 12  |-  Lim  om
4 limsssuc 6460 . . . . . . . . . . . 12  |-  ( Lim 
om  ->  ( om  C_  x  <->  om  C_  suc  x ) )
53, 4ax-mp 5 . . . . . . . . . . 11  |-  ( om  C_  x  <->  om  C_  suc  x )
6 infensuc 7485 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  om  C_  x )  ->  x  ~~  suc  x )
76ex 434 . . . . . . . . . . 11  |-  ( x  e.  On  ->  ( om  C_  x  ->  x  ~~  suc  x ) )
85, 7syl5bir 218 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( om  C_  suc  x  ->  x  ~~  suc  x ) )
92, 8sylan9r 653 . . . . . . . . 9  |-  ( ( x  e.  On  /\  ( card `  A )  =  suc  x )  -> 
( om  C_  ( card `  A )  ->  x  ~~  suc  x ) )
10 breq2 4293 . . . . . . . . . 10  |-  ( (
card `  A )  =  suc  x  ->  (
x  ~~  ( card `  A )  <->  x  ~~  suc  x ) )
1110adantl 463 . . . . . . . . 9  |-  ( ( x  e.  On  /\  ( card `  A )  =  suc  x )  -> 
( x  ~~  ( card `  A )  <->  x  ~~  suc  x ) )
129, 11sylibrd 234 . . . . . . . 8  |-  ( ( x  e.  On  /\  ( card `  A )  =  suc  x )  -> 
( om  C_  ( card `  A )  ->  x  ~~  ( card `  A
) ) )
1312ex 434 . . . . . . 7  |-  ( x  e.  On  ->  (
( card `  A )  =  suc  x  ->  ( om  C_  ( card `  A
)  ->  x  ~~  ( card `  A )
) ) )
1413com3r 79 . . . . . 6  |-  ( om  C_  ( card `  A
)  ->  ( x  e.  On  ->  ( ( card `  A )  =  suc  x  ->  x  ~~  ( card `  A
) ) ) )
1514imp 429 . . . . 5  |-  ( ( om  C_  ( card `  A )  /\  x  e.  On )  ->  (
( card `  A )  =  suc  x  ->  x  ~~  ( card `  A
) ) )
16 vex 2973 . . . . . . . . . 10  |-  x  e. 
_V
1716sucid 4794 . . . . . . . . 9  |-  x  e. 
suc  x
18 eleq2 2502 . . . . . . . . 9  |-  ( (
card `  A )  =  suc  x  ->  (
x  e.  ( card `  A )  <->  x  e.  suc  x ) )
1917, 18mpbiri 233 . . . . . . . 8  |-  ( (
card `  A )  =  suc  x  ->  x  e.  ( card `  A
) )
20 cardidm 8125 . . . . . . . 8  |-  ( card `  ( card `  A
) )  =  (
card `  A )
2119, 20syl6eleqr 2532 . . . . . . 7  |-  ( (
card `  A )  =  suc  x  ->  x  e.  ( card `  ( card `  A ) ) )
22 cardne 8131 . . . . . . 7  |-  ( x  e.  ( card `  ( card `  A ) )  ->  -.  x  ~~  ( card `  A )
)
2321, 22syl 16 . . . . . 6  |-  ( (
card `  A )  =  suc  x  ->  -.  x  ~~  ( card `  A
) )
2423a1i 11 . . . . 5  |-  ( ( om  C_  ( card `  A )  /\  x  e.  On )  ->  (
( card `  A )  =  suc  x  ->  -.  x  ~~  ( card `  A
) ) )
2515, 24pm2.65d 175 . . . 4  |-  ( ( om  C_  ( card `  A )  /\  x  e.  On )  ->  -.  ( card `  A )  =  suc  x )
2625nrexdv 2817 . . 3  |-  ( om  C_  ( card `  A
)  ->  -.  E. x  e.  On  ( card `  A
)  =  suc  x
)
27 peano1 6494 . . . . . 6  |-  (/)  e.  om
28 ssel 3347 . . . . . 6  |-  ( om  C_  ( card `  A
)  ->  ( (/)  e.  om  -> 
(/)  e.  ( card `  A ) ) )
2927, 28mpi 17 . . . . 5  |-  ( om  C_  ( card `  A
)  ->  (/)  e.  (
card `  A )
)
30 n0i 3639 . . . . 5  |-  ( (/)  e.  ( card `  A
)  ->  -.  ( card `  A )  =  (/) )
31 cardon 8110 . . . . . . . . 9  |-  ( card `  A )  e.  On
3231onordi 4819 . . . . . . . 8  |-  Ord  ( card `  A )
33 ordzsl 6455 . . . . . . . 8  |-  ( Ord  ( card `  A
)  <->  ( ( card `  A )  =  (/)  \/ 
E. x  e.  On  ( card `  A )  =  suc  x  \/  Lim  ( card `  A )
) )
3432, 33mpbi 208 . . . . . . 7  |-  ( (
card `  A )  =  (/)  \/  E. x  e.  On  ( card `  A
)  =  suc  x  \/  Lim  ( card `  A
) )
35 3orass 963 . . . . . . 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 208 . . . . . 6  |-  ( (
card `  A )  =  (/)  \/  ( E. x  e.  On  ( card `  A )  =  suc  x  \/  Lim  ( card `  A )
) )
3736ori 375 . . . . 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 377 . . 3  |-  ( om  C_  ( card `  A
)  ->  ( -.  E. x  e.  On  ( card `  A )  =  suc  x  ->  Lim  ( card `  A )
) )
4026, 39mpd 15 . 2  |-  ( om  C_  ( card `  A
)  ->  Lim  ( card `  A ) )
41 limomss 6480 . 2  |-  ( Lim  ( card `  A
)  ->  om  C_  ( card `  A ) )
4240, 41impbii 188 1  |-  ( om  C_  ( card `  A
)  <->  Lim  ( card `  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 959    = wceq 1364    e. wcel 1761   E.wrex 2714    C_ wss 3325   (/)c0 3634   class class class wbr 4289   Ord word 4714   Oncon0 4715   Lim wlim 4716   suc csuc 4717   ` cfv 5415   omcom 6475    ~~ cen 7303   cardccrd 8101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-1o 6916  df-er 7097  df-en 7307  df-dom 7308  df-card 8105
This theorem is referenced by:  infxpenlem  8176  alephislim  8249  cflim2  8428  winalim  8858  gruina  8981
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