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Theorem alephsucdom 7722
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 7716 . . 3  |-  ( B  e.  On  ->  ( aleph `  B )  ~< 
( aleph `  suc  B ) )
2 domsdomtr 7012 . . . 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 6905 . . . . 5  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~<_  ( aleph ` 
suc  B ) )
6 alephon 7712 . . . . . . 7  |-  ( aleph ` 
suc  B )  e.  On
7 ondomen 7680 . . . . . . 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 7602 . . . . . 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 6926 . . . 4  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
1311, 12syl 15 . . 3  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~~  ( card `  A ) )
14 alephnbtwn2 7715 . . . . . 6  |-  -.  (
( aleph `  B )  ~<  ( card `  A
)  /\  ( card `  A )  ~<  ( aleph `  suc  B ) )
1514imnani 412 . . . . 5  |-  ( (
aleph `  B )  ~< 
( card `  A )  ->  -.  ( card `  A
)  ~<  ( aleph `  suc  B ) )
16 ensdomtr 7013 . . . . . 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 7593 . . . . 5  |-  ( card `  A )  e.  On
20 alephon 7712 . . . . 5  |-  ( aleph `  B )  e.  On
21 domtriord 7023 . . . . 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 6935 . . 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 1696   class class class wbr 4039   Oncon0 4408   suc csuc 4410   dom cdm 4705   ` cfv 5271    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   cardccrd 7584   alephcale 7585
This theorem is referenced by:  alephsuc2  7723  alephreg  8220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589
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