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Theorem alephinit 7909
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 7906 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )
21bicomi 194 . . . 4  |-  ( A  e.  ran  aleph  <->  ( om  C_  A  /\  ( card `  A )  =  A ) )
32baib 872 . . 3  |-  ( om  C_  A  ->  ( A  e.  ran  aleph  <->  ( card `  A )  =  A ) )
43adantl 453 . 2  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  e.  ran  aleph  <->  ( card `  A )  =  A ) )
5 onenon 7769 . . . . . . . 8  |-  ( A  e.  On  ->  A  e.  dom  card )
65adantr 452 . . . . . . 7  |-  ( ( A  e.  On  /\  om  C_  A )  ->  A  e.  dom  card )
7 onenon 7769 . . . . . . 7  |-  ( x  e.  On  ->  x  e.  dom  card )
8 carddom2 7797 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  x  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  x )  <->  A  ~<_  x ) )
96, 7, 8syl2an 464 . . . . . 6  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( ( card `  A
)  C_  ( card `  x )  <->  A  ~<_  x ) )
10 cardonle 7777 . . . . . . . 8  |-  ( x  e.  On  ->  ( card `  x )  C_  x )
1110adantl 453 . . . . . . 7  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( card `  x
)  C_  x )
12 sstr 3299 . . . . . . . 8  |-  ( ( ( card `  A
)  C_  ( card `  x )  /\  ( card `  x )  C_  x )  ->  ( card `  A )  C_  x )
1312expcom 425 . . . . . . 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 227 . . . . 5  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( A  ~<_  x  -> 
( card `  A )  C_  x ) )
16 sseq1 3312 . . . . . 6  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  x  <->  A  C_  x ) )
1716imbi2d 308 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( ( A  ~<_  x  ->  ( card `  A )  C_  x )  <->  ( A  ~<_  x  ->  A  C_  x
) ) )
1815, 17syl5ibcom 212 . . . 4  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( ( card `  A
)  =  A  -> 
( A  ~<_  x  ->  A  C_  x ) ) )
1918ralrimdva 2739 . . 3  |-  ( ( A  e.  On  /\  om  C_  A )  ->  (
( card `  A )  =  A  ->  A. x  e.  On  ( A  ~<_  x  ->  A  C_  x
) ) )
20 oncardid 7776 . . . . . . 7  |-  ( A  e.  On  ->  ( card `  A )  ~~  A )
21 ensym 7092 . . . . . . 7  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
22 endom 7070 . . . . . . 7  |-  ( A 
~~  ( card `  A
)  ->  A  ~<_  ( card `  A ) )
2320, 21, 223syl 19 . . . . . 6  |-  ( A  e.  On  ->  A  ~<_  ( card `  A )
)
2423adantr 452 . . . . 5  |-  ( ( A  e.  On  /\  om  C_  A )  ->  A  ~<_  ( card `  A )
)
25 cardon 7764 . . . . . 6  |-  ( card `  A )  e.  On
26 breq2 4157 . . . . . . . 8  |-  ( x  =  ( card `  A
)  ->  ( A  ~<_  x 
<->  A  ~<_  ( card `  A
) ) )
27 sseq2 3313 . . . . . . . 8  |-  ( x  =  ( card `  A
)  ->  ( A  C_  x  <->  A  C_  ( card `  A ) ) )
2826, 27imbi12d 312 . . . . . . 7  |-  ( x  =  ( card `  A
)  ->  ( ( A  ~<_  x  ->  A  C_  x )  <->  ( A  ~<_  ( card `  A )  ->  A  C_  ( card `  A ) ) ) )
2928rspcv 2991 . . . . . 6  |-  ( (
card `  A )  e.  On  ->  ( A. x  e.  On  ( A  ~<_  x  ->  A  C_  x )  ->  ( A  ~<_  ( card `  A
)  ->  A  C_  ( card `  A ) ) ) )
3025, 29ax-mp 8 . . . . 5  |-  ( A. x  e.  On  ( A  ~<_  x  ->  A  C_  x )  ->  ( A  ~<_  ( card `  A
)  ->  A  C_  ( card `  A ) ) )
3124, 30syl5com 28 . . . 4  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x )  ->  A  C_  ( card `  A
) ) )
32 cardonle 7777 . . . . . . 7  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
3332adantr 452 . . . . . 6  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( card `  A )  C_  A )
3433biantrurd 495 . . . . 5  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  C_  ( card `  A
)  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) ) )
35 eqss 3306 . . . . 5  |-  ( (
card `  A )  =  A  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) )
3634, 35syl6bbr 255 . . . 4  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  C_  ( card `  A
)  <->  ( card `  A
)  =  A ) )
3731, 36sylibd 206 . . 3  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x )  -> 
( card `  A )  =  A ) )
3819, 37impbid 184 . 2  |-  ( ( A  e.  On  /\  om  C_  A )  ->  (
( card `  A )  =  A  <->  A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x ) ) )
394, 38bitrd 245 1  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  e.  ran  aleph  <->  A. x  e.  On  ( A  ~<_  x  ->  A  C_  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    C_ wss 3263   class class class wbr 4153   Oncon0 4522   omcom 4785   dom cdm 4818   ran crn 4819   ` cfv 5394    ~~ cen 7042    ~<_ cdom 7043   cardccrd 7755   alephcale 7756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-oi 7412  df-har 7459  df-card 7759  df-aleph 7760
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