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Theorem alephinit 8493
Description: An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
alephinit  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  e.  ran  aleph  <->  A. x  e.  On  ( A  ~<_  x  ->  A  C_  x
) ) )
Distinct variable group:    x, A

Proof of Theorem alephinit
StepHypRef Expression
1 isinfcard 8490 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )
21bicomi 202 . . . 4  |-  ( A  e.  ran  aleph  <->  ( om  C_  A  /\  ( card `  A )  =  A ) )
32baib 903 . . 3  |-  ( om  C_  A  ->  ( A  e.  ran  aleph  <->  ( card `  A )  =  A ) )
43adantl 466 . 2  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  e.  ran  aleph  <->  ( card `  A )  =  A ) )
5 onenon 8347 . . . . . . . 8  |-  ( A  e.  On  ->  A  e.  dom  card )
65adantr 465 . . . . . . 7  |-  ( ( A  e.  On  /\  om  C_  A )  ->  A  e.  dom  card )
7 onenon 8347 . . . . . . 7  |-  ( x  e.  On  ->  x  e.  dom  card )
8 carddom2 8375 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  x  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  x )  <->  A  ~<_  x ) )
96, 7, 8syl2an 477 . . . . . 6  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( ( card `  A
)  C_  ( card `  x )  <->  A  ~<_  x ) )
10 cardonle 8355 . . . . . . . 8  |-  ( x  e.  On  ->  ( card `  x )  C_  x )
1110adantl 466 . . . . . . 7  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( card `  x
)  C_  x )
12 sstr 3507 . . . . . . . 8  |-  ( ( ( card `  A
)  C_  ( card `  x )  /\  ( card `  x )  C_  x )  ->  ( card `  A )  C_  x )
1312expcom 435 . . . . . . 7  |-  ( (
card `  x )  C_  x  ->  ( ( card `  A )  C_  ( card `  x )  ->  ( card `  A
)  C_  x )
)
1411, 13syl 16 . . . . . 6  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( ( card `  A
)  C_  ( card `  x )  ->  ( card `  A )  C_  x ) )
159, 14sylbird 235 . . . . 5  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( A  ~<_  x  -> 
( card `  A )  C_  x ) )
16 sseq1 3520 . . . . . 6  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  x  <->  A  C_  x ) )
1716imbi2d 316 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( ( A  ~<_  x  ->  ( card `  A )  C_  x )  <->  ( A  ~<_  x  ->  A  C_  x
) ) )
1815, 17syl5ibcom 220 . . . 4  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( ( card `  A
)  =  A  -> 
( A  ~<_  x  ->  A  C_  x ) ) )
1918ralrimdva 2875 . . 3  |-  ( ( A  e.  On  /\  om  C_  A )  ->  (
( card `  A )  =  A  ->  A. x  e.  On  ( A  ~<_  x  ->  A  C_  x
) ) )
20 oncardid 8354 . . . . . . 7  |-  ( A  e.  On  ->  ( card `  A )  ~~  A )
21 ensym 7583 . . . . . . 7  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
22 endom 7561 . . . . . . 7  |-  ( A 
~~  ( card `  A
)  ->  A  ~<_  ( card `  A ) )
2320, 21, 223syl 20 . . . . . 6  |-  ( A  e.  On  ->  A  ~<_  ( card `  A )
)
2423adantr 465 . . . . 5  |-  ( ( A  e.  On  /\  om  C_  A )  ->  A  ~<_  ( card `  A )
)
25 cardon 8342 . . . . . 6  |-  ( card `  A )  e.  On
26 breq2 4460 . . . . . . . 8  |-  ( x  =  ( card `  A
)  ->  ( A  ~<_  x 
<->  A  ~<_  ( card `  A
) ) )
27 sseq2 3521 . . . . . . . 8  |-  ( x  =  ( card `  A
)  ->  ( A  C_  x  <->  A  C_  ( card `  A ) ) )
2826, 27imbi12d 320 . . . . . . 7  |-  ( x  =  ( card `  A
)  ->  ( ( A  ~<_  x  ->  A  C_  x )  <->  ( A  ~<_  ( card `  A )  ->  A  C_  ( card `  A ) ) ) )
2928rspcv 3206 . . . . . 6  |-  ( (
card `  A )  e.  On  ->  ( A. x  e.  On  ( A  ~<_  x  ->  A  C_  x )  ->  ( A  ~<_  ( card `  A
)  ->  A  C_  ( card `  A ) ) ) )
3025, 29ax-mp 5 . . . . 5  |-  ( A. x  e.  On  ( A  ~<_  x  ->  A  C_  x )  ->  ( A  ~<_  ( card `  A
)  ->  A  C_  ( card `  A ) ) )
3124, 30syl5com 30 . . . 4  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x )  ->  A  C_  ( card `  A
) ) )
32 cardonle 8355 . . . . . . 7  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
3332adantr 465 . . . . . 6  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( card `  A )  C_  A )
3433biantrurd 508 . . . . 5  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  C_  ( card `  A
)  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) ) )
35 eqss 3514 . . . . 5  |-  ( (
card `  A )  =  A  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) )
3634, 35syl6bbr 263 . . . 4  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  C_  ( card `  A
)  <->  ( card `  A
)  =  A ) )
3731, 36sylibd 214 . . 3  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x )  -> 
( card `  A )  =  A ) )
3819, 37impbid 191 . 2  |-  ( ( A  e.  On  /\  om  C_  A )  ->  (
( card `  A )  =  A  <->  A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x ) ) )
394, 38bitrd 253 1  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  e.  ran  aleph  <->  A. x  e.  On  ( A  ~<_  x  ->  A  C_  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   class class class wbr 4456   Oncon0 4887   dom cdm 5008   ran crn 5009   ` cfv 5594   omcom 6699    ~~ cen 7532    ~<_ cdom 7533   cardccrd 8333   alephcale 8334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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-isom 5603  df-riota 6258  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-oi 7953  df-har 8002  df-card 8337  df-aleph 8338
This theorem is referenced by: (None)
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