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Theorem alephsucdom 7893
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 7887 . . 3  |-  ( B  e.  On  ->  ( aleph `  B )  ~< 
( aleph `  suc  B ) )
2 domsdomtr 7178 . . . 4  |-  ( ( A  ~<_  ( aleph `  B
)  /\  ( aleph `  B )  ~<  ( aleph `  suc  B ) )  ->  A  ~<  (
aleph `  suc  B ) )
32ex 424 . . 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 7071 . . . . 5  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~<_  ( aleph ` 
suc  B ) )
6 alephon 7883 . . . . . 6  |-  ( aleph ` 
suc  B )  e.  On
7 ondomen 7851 . . . . . 6  |-  ( ( ( aleph `  suc  B )  e.  On  /\  A  ~<_  ( aleph `  suc  B ) )  ->  A  e.  dom  card )
86, 7mpan 652 . . . . 5  |-  ( A  ~<_  ( aleph `  suc  B )  ->  A  e.  dom  card )
9 cardid2 7773 . . . . 5  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
105, 8, 93syl 19 . . . 4  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~~  A )
1110ensymd 7094 . . 3  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~~  ( card `  A ) )
12 alephnbtwn2 7886 . . . . . 6  |-  -.  (
( aleph `  B )  ~<  ( card `  A
)  /\  ( card `  A )  ~<  ( aleph `  suc  B ) )
1312imnani 413 . . . . 5  |-  ( (
aleph `  B )  ~< 
( card `  A )  ->  -.  ( card `  A
)  ~<  ( aleph `  suc  B ) )
14 ensdomtr 7179 . . . . . 6  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~<  ( aleph `  suc  B ) )  ->  ( card `  A )  ~< 
( aleph `  suc  B ) )
1510, 14mpancom 651 . . . . 5  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~<  ( aleph `  suc  B ) )
1613, 15nsyl3 113 . . . 4  |-  ( A 
~<  ( aleph `  suc  B )  ->  -.  ( aleph `  B )  ~<  ( card `  A ) )
17 cardon 7764 . . . . 5  |-  ( card `  A )  e.  On
18 alephon 7883 . . . . 5  |-  ( aleph `  B )  e.  On
19 domtriord 7189 . . . . 5  |-  ( ( ( card `  A
)  e.  On  /\  ( aleph `  B )  e.  On )  ->  (
( card `  A )  ~<_  ( aleph `  B )  <->  -.  ( aleph `  B )  ~<  ( card `  A
) ) )
2017, 18, 19mp2an 654 . . . 4  |-  ( (
card `  A )  ~<_  ( aleph `  B )  <->  -.  ( aleph `  B )  ~<  ( card `  A
) )
2116, 20sylibr 204 . . 3  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~<_  ( aleph `  B
) )
22 endomtr 7101 . . 3  |-  ( ( A  ~~  ( card `  A )  /\  ( card `  A )  ~<_  (
aleph `  B ) )  ->  A  ~<_  ( aleph `  B ) )
2311, 21, 22syl2anc 643 . 2  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~<_  ( aleph `  B ) )
244, 23impbid1 195 1  |-  ( B  e.  On  ->  ( A  ~<_  ( aleph `  B
)  <->  A  ~<  ( aleph ` 
suc  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    e. wcel 1717   class class class wbr 4153   Oncon0 4522   suc csuc 4524   dom cdm 4818   ` cfv 5394    ~~ cen 7042    ~<_ cdom 7043    ~< csdm 7044   cardccrd 7755   alephcale 7756
This theorem is referenced by:  alephsuc2  7894  alephreg  8390
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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