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Theorem cardlim 7573
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 3175 . . . . . . . . . . 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 4643 . . . . . . . . . . . 12  |-  Lim  om
4 limsssuc 4613 . . . . . . . . . . . 12  |-  ( Lim 
om  ->  ( om  C_  x  <->  om  C_  suc  x ) )
53, 4ax-mp 10 . . . . . . . . . . 11  |-  ( om  C_  x  <->  om  C_  suc  x )
6 infensuc 7007 . . . . . . . . . . . 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 4001 . . . . . . . . . 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 2766 . . . . . . . . . 10  |-  x  e. 
_V
1716sucid 4443 . . . . . . . . 9  |-  x  e. 
suc  x
18 eleq2 2319 . . . . . . . . 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 7560 . . . . . . . 8  |-  ( card `  ( card `  A
) )  =  (
card `  A )
2119, 20syl6eleqr 2349 . . . . . . 7  |-  ( (
card `  A )  =  suc  x  ->  x  e.  ( card `  ( card `  A ) ) )
22 cardne 7566 . . . . . . 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 2621 . . 3  |-  ( om  C_  ( card `  A
)  ->  -.  E. x  e.  On  ( card `  A
)  =  suc  x
)
27 peano1 4647 . . . . . 6  |-  (/)  e.  om
28 ssel 3149 . . . . . 6  |-  ( om  C_  ( card `  A
)  ->  ( (/)  e.  om  -> 
(/)  e.  ( card `  A ) ) )
2927, 28mpi 18 . . . . 5  |-  ( om  C_  ( card `  A
)  ->  (/)  e.  (
card `  A )
)
30 n0i 3435 . . . . 5  |-  ( (/)  e.  ( card `  A
)  ->  -.  ( card `  A )  =  (/) )
31 cardon 7545 . . . . . . . . 9  |-  ( card `  A )  e.  On
3231onordi 4469 . . . . . . . 8  |-  Ord  ( card `  A )
33 ordzsl 4608 . . . . . . . 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 4633 . 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 2519    C_ wss 3127   (/)c0 3430   class class class wbr 3997   Ord word 4363   Oncon0 4364   Lim wlim 4365   suc csuc 4366   omcom 4628   ` cfv 4673    ~~ cen 6828   cardccrd 7536
This theorem is referenced by:  infxpenlem  7609  alephislim  7678  cflim2  7857  winalim  8285  gruina  8408  cartarlim  25272
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-1o 6447  df-er 6628  df-en 6832  df-dom 6833  df-card 7540
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