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Theorem alephsucdom 8492
Description: A set dominated by an aleph is strictly dominated by its successor aleph and vice-versa. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
alephsucdom  |-  ( B  e.  On  ->  ( A  ~<_  ( aleph `  B
)  <->  A  ~<  ( aleph ` 
suc  B ) ) )

Proof of Theorem alephsucdom
StepHypRef Expression
1 alephordilem1 8486 . . 3  |-  ( B  e.  On  ->  ( aleph `  B )  ~< 
( aleph `  suc  B ) )
2 domsdomtr 7690 . . . 4  |-  ( ( A  ~<_  ( aleph `  B
)  /\  ( aleph `  B )  ~<  ( aleph `  suc  B ) )  ->  A  ~<  (
aleph `  suc  B ) )
32ex 432 . . 3  |-  ( A  ~<_  ( aleph `  B )  ->  ( ( aleph `  B
)  ~<  ( aleph `  suc  B )  ->  A  ~<  (
aleph `  suc  B ) ) )
41, 3syl5com 28 . 2  |-  ( B  e.  On  ->  ( A  ~<_  ( aleph `  B
)  ->  A  ~<  (
aleph `  suc  B ) ) )
5 sdomdom 7581 . . . . 5  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~<_  ( aleph ` 
suc  B ) )
6 alephon 8482 . . . . . 6  |-  ( aleph ` 
suc  B )  e.  On
7 ondomen 8450 . . . . . 6  |-  ( ( ( aleph `  suc  B )  e.  On  /\  A  ~<_  ( aleph `  suc  B ) )  ->  A  e.  dom  card )
86, 7mpan 668 . . . . 5  |-  ( A  ~<_  ( aleph `  suc  B )  ->  A  e.  dom  card )
9 cardid2 8366 . . . . 5  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
105, 8, 93syl 18 . . . 4  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~~  A )
1110ensymd 7604 . . 3  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~~  ( card `  A ) )
12 alephnbtwn2 8485 . . . . . 6  |-  -.  (
( aleph `  B )  ~<  ( card `  A
)  /\  ( card `  A )  ~<  ( aleph `  suc  B ) )
1312imnani 421 . . . . 5  |-  ( (
aleph `  B )  ~< 
( card `  A )  ->  -.  ( card `  A
)  ~<  ( aleph `  suc  B ) )
14 ensdomtr 7691 . . . . . 6  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~<  ( aleph `  suc  B ) )  ->  ( card `  A )  ~< 
( aleph `  suc  B ) )
1510, 14mpancom 667 . . . . 5  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~<  ( aleph `  suc  B ) )
1613, 15nsyl3 119 . . . 4  |-  ( A 
~<  ( aleph `  suc  B )  ->  -.  ( aleph `  B )  ~<  ( card `  A ) )
17 cardon 8357 . . . . 5  |-  ( card `  A )  e.  On
18 alephon 8482 . . . . 5  |-  ( aleph `  B )  e.  On
19 domtriord 7701 . . . . 5  |-  ( ( ( card `  A
)  e.  On  /\  ( aleph `  B )  e.  On )  ->  (
( card `  A )  ~<_  ( aleph `  B )  <->  -.  ( aleph `  B )  ~<  ( card `  A
) ) )
2017, 18, 19mp2an 670 . . . 4  |-  ( (
card `  A )  ~<_  ( aleph `  B )  <->  -.  ( aleph `  B )  ~<  ( card `  A
) )
2116, 20sylibr 212 . . 3  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~<_  ( aleph `  B
) )
22 endomtr 7611 . . 3  |-  ( ( A  ~~  ( card `  A )  /\  ( card `  A )  ~<_  (
aleph `  B ) )  ->  A  ~<_  ( aleph `  B ) )
2311, 21, 22syl2anc 659 . 2  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~<_  ( aleph `  B ) )
244, 23impbid1 203 1  |-  ( B  e.  On  ->  ( A  ~<_  ( aleph `  B
)  <->  A  ~<  ( aleph ` 
suc  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    e. wcel 1842   class class class wbr 4395   dom cdm 4823   Oncon0 5410   suc csuc 5412   ` cfv 5569    ~~ cen 7551    ~<_ cdom 7552    ~< csdm 7553   cardccrd 8348   alephcale 8349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-oi 7969  df-har 8018  df-card 8352  df-aleph 8353
This theorem is referenced by:  alephsuc2  8493  alephreg  8989
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