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Theorem alephsucdom 8350
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 8344 . . 3  |-  ( B  e.  On  ->  ( aleph `  B )  ~< 
( aleph `  suc  B ) )
2 domsdomtr 7546 . . . 4  |-  ( ( A  ~<_  ( aleph `  B
)  /\  ( aleph `  B )  ~<  ( aleph `  suc  B ) )  ->  A  ~<  (
aleph `  suc  B ) )
32ex 434 . . 3  |-  ( A  ~<_  ( aleph `  B )  ->  ( ( aleph `  B
)  ~<  ( aleph `  suc  B )  ->  A  ~<  (
aleph `  suc  B ) ) )
41, 3syl5com 30 . 2  |-  ( B  e.  On  ->  ( A  ~<_  ( aleph `  B
)  ->  A  ~<  (
aleph `  suc  B ) ) )
5 sdomdom 7437 . . . . 5  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~<_  ( aleph ` 
suc  B ) )
6 alephon 8340 . . . . . 6  |-  ( aleph ` 
suc  B )  e.  On
7 ondomen 8308 . . . . . 6  |-  ( ( ( aleph `  suc  B )  e.  On  /\  A  ~<_  ( aleph `  suc  B ) )  ->  A  e.  dom  card )
86, 7mpan 670 . . . . 5  |-  ( A  ~<_  ( aleph `  suc  B )  ->  A  e.  dom  card )
9 cardid2 8224 . . . . 5  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
105, 8, 93syl 20 . . . 4  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~~  A )
1110ensymd 7460 . . 3  |-  ( A 
~<  ( aleph `  suc  B )  ->  A  ~~  ( card `  A ) )
12 alephnbtwn2 8343 . . . . . 6  |-  -.  (
( aleph `  B )  ~<  ( card `  A
)  /\  ( card `  A )  ~<  ( aleph `  suc  B ) )
1312imnani 423 . . . . 5  |-  ( (
aleph `  B )  ~< 
( card `  A )  ->  -.  ( card `  A
)  ~<  ( aleph `  suc  B ) )
14 ensdomtr 7547 . . . . . 6  |-  ( ( ( card `  A
)  ~~  A  /\  A  ~<  ( aleph `  suc  B ) )  ->  ( card `  A )  ~< 
( aleph `  suc  B ) )
1510, 14mpancom 669 . . . . 5  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~<  ( aleph `  suc  B ) )
1613, 15nsyl3 119 . . . 4  |-  ( A 
~<  ( aleph `  suc  B )  ->  -.  ( aleph `  B )  ~<  ( card `  A ) )
17 cardon 8215 . . . . 5  |-  ( card `  A )  e.  On
18 alephon 8340 . . . . 5  |-  ( aleph `  B )  e.  On
19 domtriord 7557 . . . . 5  |-  ( ( ( card `  A
)  e.  On  /\  ( aleph `  B )  e.  On )  ->  (
( card `  A )  ~<_  ( aleph `  B )  <->  -.  ( aleph `  B )  ~<  ( card `  A
) ) )
2017, 18, 19mp2an 672 . . . 4  |-  ( (
card `  A )  ~<_  ( aleph `  B )  <->  -.  ( aleph `  B )  ~<  ( card `  A
) )
2116, 20sylibr 212 . . 3  |-  ( A 
~<  ( aleph `  suc  B )  ->  ( card `  A
)  ~<_  ( aleph `  B
) )
22 endomtr 7467 . . 3  |-  ( ( A  ~~  ( card `  A )  /\  ( card `  A )  ~<_  (
aleph `  B ) )  ->  A  ~<_  ( aleph `  B ) )
2311, 21, 22syl2anc 661 . 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 1758   class class class wbr 4390   Oncon0 4817   suc csuc 4819   dom cdm 4938   ` cfv 5516    ~~ cen 7407    ~<_ cdom 7408    ~< csdm 7409   cardccrd 8206   alephcale 8207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-oi 7825  df-har 7874  df-card 8210  df-aleph 8211
This theorem is referenced by:  alephsuc2  8351  alephreg  8847
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