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Theorem alephinit 7968
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 7965 . . . . 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 7828 . . . . . . . 8  |-  ( A  e.  On  ->  A  e.  dom  card )
65adantr 452 . . . . . . 7  |-  ( ( A  e.  On  /\  om  C_  A )  ->  A  e.  dom  card )
7 onenon 7828 . . . . . . 7  |-  ( x  e.  On  ->  x  e.  dom  card )
8 carddom2 7856 . . . . . . 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 7836 . . . . . . . 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 3348 . . . . . . . 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 3361 . . . . . 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 2788 . . 3  |-  ( ( A  e.  On  /\  om  C_  A )  ->  (
( card `  A )  =  A  ->  A. x  e.  On  ( A  ~<_  x  ->  A  C_  x
) ) )
20 oncardid 7835 . . . . . . 7  |-  ( A  e.  On  ->  ( card `  A )  ~~  A )
21 ensym 7148 . . . . . . 7  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
22 endom 7126 . . . . . . 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 7823 . . . . . 6  |-  ( card `  A )  e.  On
26 breq2 4208 . . . . . . . 8  |-  ( x  =  ( card `  A
)  ->  ( A  ~<_  x 
<->  A  ~<_  ( card `  A
) ) )
27 sseq2 3362 . . . . . . . 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 3040 . . . . . 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 7836 . . . . . . 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 3355 . . . . 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 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   class class class wbr 4204   Oncon0 4573   omcom 4837   dom cdm 4870   ran crn 4871   ` cfv 5446    ~~ cen 7098    ~<_ cdom 7099   cardccrd 7814   alephcale 7815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-har 7518  df-card 7818  df-aleph 7819
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