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Theorem alephnbtwn2 7699
Description: No set has equinumerosity between an aleph and its successor aleph. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
alephnbtwn2  |-  -.  (
( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )

Proof of Theorem alephnbtwn2
StepHypRef Expression
1 cardidm 7592 . . 3  |-  ( card `  ( card `  B
) )  =  (
card `  B )
2 alephnbtwn 7698 . . 3  |-  ( (
card `  ( card `  B ) )  =  ( card `  B
)  ->  -.  (
( aleph `  A )  e.  ( card `  B
)  /\  ( card `  B )  e.  (
aleph `  suc  A ) ) )
31, 2ax-mp 8 . 2  |-  -.  (
( aleph `  A )  e.  ( card `  B
)  /\  ( card `  B )  e.  (
aleph `  suc  A ) )
4 alephon 7696 . . . . . . . 8  |-  ( aleph ` 
suc  A )  e.  On
5 sdomdom 6889 . . . . . . . 8  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  ~<_  ( aleph ` 
suc  A ) )
6 ondomen 7664 . . . . . . . 8  |-  ( ( ( aleph `  suc  A )  e.  On  /\  B  ~<_  ( aleph `  suc  A ) )  ->  B  e.  dom  card )
74, 5, 6sylancr 644 . . . . . . 7  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  e.  dom  card )
8 cardid2 7586 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
97, 8syl 15 . . . . . 6  |-  ( B 
~<  ( aleph `  suc  A )  ->  ( card `  B
)  ~~  B )
10 ensym 6910 . . . . . 6  |-  ( (
card `  B )  ~~  B  ->  B  ~~  ( card `  B )
)
119, 10syl 15 . . . . 5  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  ~~  ( card `  B ) )
12 sdomentr 6995 . . . . 5  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~~  ( card `  B )
)  ->  ( aleph `  A )  ~<  ( card `  B ) )
1311, 12sylan2 460 . . . 4  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( aleph `  A )  ~<  ( card `  B ) )
14 alephon 7696 . . . . . 6  |-  ( aleph `  A )  e.  On
15 cardon 7577 . . . . . . 7  |-  ( card `  B )  e.  On
16 onenon 7582 . . . . . . 7  |-  ( (
card `  B )  e.  On  ->  ( card `  B )  e.  dom  card )
1715, 16ax-mp 8 . . . . . 6  |-  ( card `  B )  e.  dom  card
18 cardsdomel 7607 . . . . . 6  |-  ( ( ( aleph `  A )  e.  On  /\  ( card `  B )  e.  dom  card )  ->  ( ( aleph `  A )  ~< 
( card `  B )  <->  (
aleph `  A )  e.  ( card `  ( card `  B ) ) ) )
1914, 17, 18mp2an 653 . . . . 5  |-  ( (
aleph `  A )  ~< 
( card `  B )  <->  (
aleph `  A )  e.  ( card `  ( card `  B ) ) )
201eleq2i 2347 . . . . 5  |-  ( (
aleph `  A )  e.  ( card `  ( card `  B ) )  <-> 
( aleph `  A )  e.  ( card `  B
) )
2119, 20bitri 240 . . . 4  |-  ( (
aleph `  A )  ~< 
( card `  B )  <->  (
aleph `  A )  e.  ( card `  B
) )
2213, 21sylib 188 . . 3  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( aleph `  A )  e.  (
card `  B )
)
23 ensdomtr 6997 . . . . . 6  |-  ( ( ( card `  B
)  ~~  B  /\  B  ~<  ( aleph `  suc  A ) )  ->  ( card `  B )  ~< 
( aleph `  suc  A ) )
249, 23mpancom 650 . . . . 5  |-  ( B 
~<  ( aleph `  suc  A )  ->  ( card `  B
)  ~<  ( aleph `  suc  A ) )
2524adantl 452 . . . 4  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( card `  B )  ~<  ( aleph `  suc  A ) )
26 onenon 7582 . . . . . . 7  |-  ( (
aleph `  suc  A )  e.  On  ->  ( aleph `  suc  A )  e.  dom  card )
274, 26ax-mp 8 . . . . . 6  |-  ( aleph ` 
suc  A )  e. 
dom  card
28 cardsdomel 7607 . . . . . 6  |-  ( ( ( card `  B
)  e.  On  /\  ( aleph `  suc  A )  e.  dom  card )  ->  ( ( card `  B
)  ~<  ( aleph `  suc  A )  <->  ( card `  B
)  e.  ( card `  ( aleph `  suc  A ) ) ) )
2915, 27, 28mp2an 653 . . . . 5  |-  ( (
card `  B )  ~<  ( aleph `  suc  A )  <-> 
( card `  B )  e.  ( card `  ( aleph `  suc  A ) ) )
30 alephcard 7697 . . . . . 6  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
3130eleq2i 2347 . . . . 5  |-  ( (
card `  B )  e.  ( card `  ( aleph `  suc  A ) )  <->  ( card `  B
)  e.  ( aleph ` 
suc  A ) )
3229, 31bitri 240 . . . 4  |-  ( (
card `  B )  ~<  ( aleph `  suc  A )  <-> 
( card `  B )  e.  ( aleph `  suc  A ) )
3325, 32sylib 188 . . 3  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( card `  B )  e.  (
aleph `  suc  A ) )
3422, 33jca 518 . 2  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( ( aleph `  A )  e.  ( card `  B
)  /\  ( card `  B )  e.  (
aleph `  suc  A ) ) )
353, 34mto 167 1  |-  -.  (
( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    = wceq 1623    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:  alephsucdom  7706  alephsucpw2  7738  alephgch  8300  winalim2  8318  aleph1re  12523
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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