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Theorem alephsucdom 7706
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 7700 . . 3  |-  ( B  e.  On  ->  ( aleph `  B )  ~< 
( aleph `  suc  B ) )
2 domsdomtr 6996 . . . 4  |-  ( ( A  ~<_  ( aleph `  B
)  /\  ( aleph `  B )  ~<  ( aleph `  suc  B ) )  ->  A  ~<  (
aleph `  suc  B ) )
32ex 423 . . 3  |-  ( A  ~<_  ( aleph `  B )  ->  ( ( aleph `  B
)  ~<  ( aleph `  suc  B )  ->  A  ~<  (
aleph `  suc  B ) ) )
41, 3syl5com 26 . 2  |-  ( B  e.  On  ->  ( A  ~<_  ( aleph `  B
)  ->  A  ~<  (
aleph `  suc  B ) ) )
5 sdomdom 6889 . . . . 5  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~<_  ( aleph ` 
suc  B ) )
6 alephon 7696 . . . . . . 7  |-  ( aleph ` 
suc  B )  e.  On
7 ondomen 7664 . . . . . . 7  |-  ( ( ( aleph `  suc  B )  e.  On  /\  A  ~<_  ( aleph `  suc  B ) )  ->  A  e.  dom  card )
86, 7mpan 651 . . . . . 6  |-  ( A  ~<_  ( aleph `  suc  B )  ->  A  e.  dom  card )
9 cardid2 7586 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
108, 9syl 15 . . . . 5  |-  ( A  ~<_  ( aleph `  suc  B )  ->  ( card `  A
)  ~~  A )
115, 10syl 15 . . . 4  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~~  A )
12 ensym 6910 . . . 4  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
1311, 12syl 15 . . 3  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~~  ( card `  A ) )
14 alephnbtwn2 7699 . . . . . 6  |-  -.  (
( aleph `  B )  ~<  ( card `  A
)  /\  ( card `  A )  ~<  ( aleph `  suc  B ) )
1514imnani 412 . . . . 5  |-  ( (
aleph `  B )  ~< 
( card `  A )  ->  -.  ( card `  A
)  ~<  ( aleph `  suc  B ) )
16 ensdomtr 6997 . . . . . 6  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~<  ( aleph `  suc  B ) )  ->  ( card `  A )  ~< 
( aleph `  suc  B ) )
1711, 16mpancom 650 . . . . 5  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~<  ( aleph `  suc  B ) )
1815, 17nsyl3 111 . . . 4  |-  ( A 
~<  ( aleph `  suc  B )  ->  -.  ( aleph `  B )  ~<  ( card `  A ) )
19 cardon 7577 . . . . 5  |-  ( card `  A )  e.  On
20 alephon 7696 . . . . 5  |-  ( aleph `  B )  e.  On
21 domtriord 7007 . . . . 5  |-  ( ( ( card `  A
)  e.  On  /\  ( aleph `  B )  e.  On )  ->  (
( card `  A )  ~<_  ( aleph `  B )  <->  -.  ( aleph `  B )  ~<  ( card `  A
) ) )
2219, 20, 21mp2an 653 . . . 4  |-  ( (
card `  A )  ~<_  ( aleph `  B )  <->  -.  ( aleph `  B )  ~<  ( card `  A
) )
2318, 22sylibr 203 . . 3  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~<_  ( aleph `  B
) )
24 endomtr 6919 . . 3  |-  ( ( A  ~~  ( card `  A )  /\  ( card `  A )  ~<_  (
aleph `  B ) )  ->  A  ~<_  ( aleph `  B ) )
2513, 23, 24syl2anc 642 . 2  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~<_  ( aleph `  B ) )
264, 25impbid1 194 1  |-  ( B  e.  On  ->  ( A  ~<_  ( aleph `  B
)  <->  A  ~<  ( aleph ` 
suc  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    e. wcel 1684   class class class wbr 4023   Oncon0 4392   suc csuc 4394   dom cdm 4689   ` cfv 5255    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568   alephcale 7569
This theorem is referenced by:  alephsuc2  7707  alephreg  8204
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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