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Theorem alephinit 7722
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 7719 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )
21bicomi 193 . . . 4  |-  ( A  e.  ran  aleph  <->  ( om  C_  A  /\  ( card `  A )  =  A ) )
32baib 871 . . 3  |-  ( om  C_  A  ->  ( A  e.  ran  aleph  <->  ( card `  A )  =  A ) )
43adantl 452 . 2  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  e.  ran  aleph  <->  ( card `  A )  =  A ) )
5 onenon 7582 . . . . . . . 8  |-  ( A  e.  On  ->  A  e.  dom  card )
65adantr 451 . . . . . . 7  |-  ( ( A  e.  On  /\  om  C_  A )  ->  A  e.  dom  card )
7 onenon 7582 . . . . . . 7  |-  ( x  e.  On  ->  x  e.  dom  card )
8 carddom2 7610 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  x  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  x )  <->  A  ~<_  x ) )
96, 7, 8syl2an 463 . . . . . 6  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( ( card `  A
)  C_  ( card `  x )  <->  A  ~<_  x ) )
10 cardonle 7590 . . . . . . . 8  |-  ( x  e.  On  ->  ( card `  x )  C_  x )
1110adantl 452 . . . . . . 7  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( card `  x
)  C_  x )
12 sstr 3187 . . . . . . . 8  |-  ( ( ( card `  A
)  C_  ( card `  x )  /\  ( card `  x )  C_  x )  ->  ( card `  A )  C_  x )
1312expcom 424 . . . . . . 7  |-  ( (
card `  x )  C_  x  ->  ( ( card `  A )  C_  ( card `  x )  ->  ( card `  A
)  C_  x )
)
1411, 13syl 15 . . . . . 6  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( ( card `  A
)  C_  ( card `  x )  ->  ( card `  A )  C_  x ) )
159, 14sylbird 226 . . . . 5  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( A  ~<_  x  -> 
( card `  A )  C_  x ) )
16 sseq1 3199 . . . . . 6  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  x  <->  A  C_  x ) )
1716imbi2d 307 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( ( A  ~<_  x  ->  ( card `  A )  C_  x )  <->  ( A  ~<_  x  ->  A  C_  x
) ) )
1815, 17syl5ibcom 211 . . . 4  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( ( card `  A
)  =  A  -> 
( A  ~<_  x  ->  A  C_  x ) ) )
1918ralrimdva 2633 . . 3  |-  ( ( A  e.  On  /\  om  C_  A )  ->  (
( card `  A )  =  A  ->  A. x  e.  On  ( A  ~<_  x  ->  A  C_  x
) ) )
20 oncardid 7589 . . . . . . 7  |-  ( A  e.  On  ->  ( card `  A )  ~~  A )
21 ensym 6910 . . . . . . 7  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
22 endom 6888 . . . . . . 7  |-  ( A 
~~  ( card `  A
)  ->  A  ~<_  ( card `  A ) )
2320, 21, 223syl 18 . . . . . 6  |-  ( A  e.  On  ->  A  ~<_  ( card `  A )
)
2423adantr 451 . . . . 5  |-  ( ( A  e.  On  /\  om  C_  A )  ->  A  ~<_  ( card `  A )
)
25 cardon 7577 . . . . . 6  |-  ( card `  A )  e.  On
26 breq2 4027 . . . . . . . 8  |-  ( x  =  ( card `  A
)  ->  ( A  ~<_  x 
<->  A  ~<_  ( card `  A
) ) )
27 sseq2 3200 . . . . . . . 8  |-  ( x  =  ( card `  A
)  ->  ( A  C_  x  <->  A  C_  ( card `  A ) ) )
2826, 27imbi12d 311 . . . . . . 7  |-  ( x  =  ( card `  A
)  ->  ( ( A  ~<_  x  ->  A  C_  x )  <->  ( A  ~<_  ( card `  A )  ->  A  C_  ( card `  A ) ) ) )
2928rspcv 2880 . . . . . 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 26 . . . 4  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x )  ->  A  C_  ( card `  A
) ) )
32 cardonle 7590 . . . . . . 7  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
3332adantr 451 . . . . . 6  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( card `  A )  C_  A )
3433biantrurd 494 . . . . 5  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  C_  ( card `  A
)  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) ) )
35 eqss 3194 . . . . 5  |-  ( (
card `  A )  =  A  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) )
3634, 35syl6bbr 254 . . . 4  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  C_  ( card `  A
)  <->  ( card `  A
)  =  A ) )
3731, 36sylibd 205 . . 3  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x )  -> 
( card `  A )  =  A ) )
3819, 37impbid 183 . 2  |-  ( ( A  e.  On  /\  om  C_  A )  ->  (
( card `  A )  =  A  <->  A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x ) ) )
394, 38bitrd 244 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023   Oncon0 4392   omcom 4656   dom cdm 4689   ran crn 4690   ` cfv 5255    ~~ cen 6860    ~<_ cdom 6861   cardccrd 7568   alephcale 7569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573
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