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Theorem alephinit 8523
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 8520 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  <->  A  e.  ran  aleph )
21bicomi 206 . . . 4  |-  ( A  e.  ran  aleph  <->  ( om  C_  A  /\  ( card `  A )  =  A ) )
32baib 913 . . 3  |-  ( om  C_  A  ->  ( A  e.  ran  aleph  <->  ( card `  A )  =  A ) )
43adantl 468 . 2  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  e.  ran  aleph  <->  ( card `  A )  =  A ) )
5 onenon 8380 . . . . . . . 8  |-  ( A  e.  On  ->  A  e.  dom  card )
65adantr 467 . . . . . . 7  |-  ( ( A  e.  On  /\  om  C_  A )  ->  A  e.  dom  card )
7 onenon 8380 . . . . . . 7  |-  ( x  e.  On  ->  x  e.  dom  card )
8 carddom2 8408 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  x  e.  dom  card )  ->  ( ( card `  A )  C_  ( card `  x )  <->  A  ~<_  x ) )
96, 7, 8syl2an 480 . . . . . 6  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( ( card `  A
)  C_  ( card `  x )  <->  A  ~<_  x ) )
10 cardonle 8388 . . . . . . . 8  |-  ( x  e.  On  ->  ( card `  x )  C_  x )
1110adantl 468 . . . . . . 7  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( card `  x
)  C_  x )
12 sstr 3439 . . . . . . . 8  |-  ( ( ( card `  A
)  C_  ( card `  x )  /\  ( card `  x )  C_  x )  ->  ( card `  A )  C_  x )
1312expcom 437 . . . . . . 7  |-  ( (
card `  x )  C_  x  ->  ( ( card `  A )  C_  ( card `  x )  ->  ( card `  A
)  C_  x )
)
1411, 13syl 17 . . . . . 6  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( ( card `  A
)  C_  ( card `  x )  ->  ( card `  A )  C_  x ) )
159, 14sylbird 239 . . . . 5  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( A  ~<_  x  -> 
( card `  A )  C_  x ) )
16 sseq1 3452 . . . . . 6  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  C_  x  <->  A  C_  x ) )
1716imbi2d 318 . . . . 5  |-  ( (
card `  A )  =  A  ->  ( ( A  ~<_  x  ->  ( card `  A )  C_  x )  <->  ( A  ~<_  x  ->  A  C_  x
) ) )
1815, 17syl5ibcom 224 . . . 4  |-  ( ( ( A  e.  On  /\ 
om  C_  A )  /\  x  e.  On )  ->  ( ( card `  A
)  =  A  -> 
( A  ~<_  x  ->  A  C_  x ) ) )
1918ralrimdva 2805 . . 3  |-  ( ( A  e.  On  /\  om  C_  A )  ->  (
( card `  A )  =  A  ->  A. x  e.  On  ( A  ~<_  x  ->  A  C_  x
) ) )
20 oncardid 8387 . . . . . . 7  |-  ( A  e.  On  ->  ( card `  A )  ~~  A )
21 ensym 7615 . . . . . . 7  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
22 endom 7593 . . . . . . 7  |-  ( A 
~~  ( card `  A
)  ->  A  ~<_  ( card `  A ) )
2320, 21, 223syl 18 . . . . . 6  |-  ( A  e.  On  ->  A  ~<_  ( card `  A )
)
2423adantr 467 . . . . 5  |-  ( ( A  e.  On  /\  om  C_  A )  ->  A  ~<_  ( card `  A )
)
25 cardon 8375 . . . . . 6  |-  ( card `  A )  e.  On
26 breq2 4405 . . . . . . . 8  |-  ( x  =  ( card `  A
)  ->  ( A  ~<_  x 
<->  A  ~<_  ( card `  A
) ) )
27 sseq2 3453 . . . . . . . 8  |-  ( x  =  ( card `  A
)  ->  ( A  C_  x  <->  A  C_  ( card `  A ) ) )
2826, 27imbi12d 322 . . . . . . 7  |-  ( x  =  ( card `  A
)  ->  ( ( A  ~<_  x  ->  A  C_  x )  <->  ( A  ~<_  ( card `  A )  ->  A  C_  ( card `  A ) ) ) )
2928rspcv 3145 . . . . . 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 31 . . . 4  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x )  ->  A  C_  ( card `  A
) ) )
32 cardonle 8388 . . . . . . 7  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
3332adantr 467 . . . . . 6  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( card `  A )  C_  A )
3433biantrurd 511 . . . . 5  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  C_  ( card `  A
)  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) ) )
35 eqss 3446 . . . . 5  |-  ( (
card `  A )  =  A  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) )
3634, 35syl6bbr 267 . . . 4  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  C_  ( card `  A
)  <->  ( card `  A
)  =  A ) )
3731, 36sylibd 218 . . 3  |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x )  -> 
( card `  A )  =  A ) )
3819, 37impbid 194 . 2  |-  ( ( A  e.  On  /\  om  C_  A )  ->  (
( card `  A )  =  A  <->  A. x  e.  On  ( A  ~<_  x  ->  A 
C_  x ) ) )
394, 38bitrd 257 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 188    /\ wa 371    = wceq 1443    e. wcel 1886   A.wral 2736    C_ wss 3403   class class class wbr 4401   dom cdm 4833   ran crn 4834   Oncon0 5422   ` cfv 5581   omcom 6689    ~~ cen 7563    ~<_ cdom 7564   cardccrd 8366   alephcale 8367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-om 6690  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-oi 8022  df-har 8070  df-card 8370  df-aleph 8371
This theorem is referenced by: (None)
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