MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardlim Structured version   Unicode version

Theorem cardlim 8365
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 3531 . . . . . . . . . . 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 6710 . . . . . . . . . . . 12  |-  Lim  om
4 limsssuc 6680 . . . . . . . . . . . 12  |-  ( Lim 
om  ->  ( om  C_  x  <->  om  C_  suc  x ) )
53, 4ax-mp 5 . . . . . . . . . . 11  |-  ( om  C_  x  <->  om  C_  suc  x )
6 infensuc 7707 . . . . . . . . . . . 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 658 . . . . . . . . 9  |-  ( ( x  e.  On  /\  ( card `  A )  =  suc  x )  -> 
( om  C_  ( card `  A )  ->  x  ~~  suc  x ) )
10 breq2 4457 . . . . . . . . . 10  |-  ( (
card `  A )  =  suc  x  ->  (
x  ~~  ( card `  A )  <->  x  ~~  suc  x ) )
1110adantl 466 . . . . . . . . 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 3121 . . . . . . . . . 10  |-  x  e. 
_V
1716sucid 4963 . . . . . . . . 9  |-  x  e. 
suc  x
18 eleq2 2540 . . . . . . . . 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 8352 . . . . . . . 8  |-  ( card `  ( card `  A
) )  =  (
card `  A )
2119, 20syl6eleqr 2566 . . . . . . 7  |-  ( (
card `  A )  =  suc  x  ->  x  e.  ( card `  ( card `  A ) ) )
22 cardne 8358 . . . . . . 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 2923 . . 3  |-  ( om  C_  ( card `  A
)  ->  -.  E. x  e.  On  ( card `  A
)  =  suc  x
)
27 peano1 6714 . . . . . 6  |-  (/)  e.  om
28 ssel 3503 . . . . . 6  |-  ( om  C_  ( card `  A
)  ->  ( (/)  e.  om  -> 
(/)  e.  ( card `  A ) ) )
2927, 28mpi 17 . . . . 5  |-  ( om  C_  ( card `  A
)  ->  (/)  e.  (
card `  A )
)
30 n0i 3795 . . . . 5  |-  ( (/)  e.  ( card `  A
)  ->  -.  ( card `  A )  =  (/) )
31 cardon 8337 . . . . . . . . 9  |-  ( card `  A )  e.  On
3231onordi 4988 . . . . . . . 8  |-  Ord  ( card `  A )
33 ordzsl 6675 . . . . . . . 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 976 . . . . . . 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 6700 . 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 972    = wceq 1379    e. wcel 1767   E.wrex 2818    C_ wss 3481   (/)c0 3790   class class class wbr 4453   Ord word 4883   Oncon0 4884   Lim wlim 4885   suc csuc 4886   ` cfv 5594   omcom 6695    ~~ cen 7525   cardccrd 8328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-card 8332
This theorem is referenced by:  infxpenlem  8403  alephislim  8476  cflim2  8655  winalim  9085  gruina  9208
  Copyright terms: Public domain W3C validator