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Theorem alephnbtwn2 7887
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 7780 . . 3  |-  ( card `  ( card `  B
) )  =  (
card `  B )
2 alephnbtwn 7886 . . 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 7884 . . . . . . . 8  |-  ( aleph ` 
suc  A )  e.  On
5 sdomdom 7072 . . . . . . . 8  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  ~<_  ( aleph ` 
suc  A ) )
6 ondomen 7852 . . . . . . . 8  |-  ( ( ( aleph `  suc  A )  e.  On  /\  B  ~<_  ( aleph `  suc  A ) )  ->  B  e.  dom  card )
74, 5, 6sylancr 645 . . . . . . 7  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  e.  dom  card )
8 cardid2 7774 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
97, 8syl 16 . . . . . 6  |-  ( B 
~<  ( aleph `  suc  A )  ->  ( card `  B
)  ~~  B )
109ensymd 7095 . . . . 5  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  ~~  ( card `  B ) )
11 sdomentr 7178 . . . . 5  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~~  ( card `  B )
)  ->  ( aleph `  A )  ~<  ( card `  B ) )
1210, 11sylan2 461 . . . 4  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( aleph `  A )  ~<  ( card `  B ) )
13 alephon 7884 . . . . . 6  |-  ( aleph `  A )  e.  On
14 cardon 7765 . . . . . . 7  |-  ( card `  B )  e.  On
15 onenon 7770 . . . . . . 7  |-  ( (
card `  B )  e.  On  ->  ( card `  B )  e.  dom  card )
1614, 15ax-mp 8 . . . . . 6  |-  ( card `  B )  e.  dom  card
17 cardsdomel 7795 . . . . . 6  |-  ( ( ( aleph `  A )  e.  On  /\  ( card `  B )  e.  dom  card )  ->  ( ( aleph `  A )  ~< 
( card `  B )  <->  (
aleph `  A )  e.  ( card `  ( card `  B ) ) ) )
1813, 16, 17mp2an 654 . . . . 5  |-  ( (
aleph `  A )  ~< 
( card `  B )  <->  (
aleph `  A )  e.  ( card `  ( card `  B ) ) )
191eleq2i 2452 . . . . 5  |-  ( (
aleph `  A )  e.  ( card `  ( card `  B ) )  <-> 
( aleph `  A )  e.  ( card `  B
) )
2018, 19bitri 241 . . . 4  |-  ( (
aleph `  A )  ~< 
( card `  B )  <->  (
aleph `  A )  e.  ( card `  B
) )
2112, 20sylib 189 . . 3  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( aleph `  A )  e.  (
card `  B )
)
22 ensdomtr 7180 . . . . . 6  |-  ( ( ( card `  B
)  ~~  B  /\  B  ~<  ( aleph `  suc  A ) )  ->  ( card `  B )  ~< 
( aleph `  suc  A ) )
239, 22mpancom 651 . . . . 5  |-  ( B 
~<  ( aleph `  suc  A )  ->  ( card `  B
)  ~<  ( aleph `  suc  A ) )
2423adantl 453 . . . 4  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( card `  B )  ~<  ( aleph `  suc  A ) )
25 onenon 7770 . . . . . . 7  |-  ( (
aleph `  suc  A )  e.  On  ->  ( aleph `  suc  A )  e.  dom  card )
264, 25ax-mp 8 . . . . . 6  |-  ( aleph ` 
suc  A )  e. 
dom  card
27 cardsdomel 7795 . . . . . 6  |-  ( ( ( card `  B
)  e.  On  /\  ( aleph `  suc  A )  e.  dom  card )  ->  ( ( card `  B
)  ~<  ( aleph `  suc  A )  <->  ( card `  B
)  e.  ( card `  ( aleph `  suc  A ) ) ) )
2814, 26, 27mp2an 654 . . . . 5  |-  ( (
card `  B )  ~<  ( aleph `  suc  A )  <-> 
( card `  B )  e.  ( card `  ( aleph `  suc  A ) ) )
29 alephcard 7885 . . . . . 6  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
3029eleq2i 2452 . . . . 5  |-  ( (
card `  B )  e.  ( card `  ( aleph `  suc  A ) )  <->  ( card `  B
)  e.  ( aleph ` 
suc  A ) )
3128, 30bitri 241 . . . 4  |-  ( (
card `  B )  ~<  ( aleph `  suc  A )  <-> 
( card `  B )  e.  ( aleph `  suc  A ) )
3224, 31sylib 189 . . 3  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( card `  B )  e.  (
aleph `  suc  A ) )
3321, 32jca 519 . 2  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( ( aleph `  A )  e.  ( card `  B
)  /\  ( card `  B )  e.  (
aleph `  suc  A ) ) )
343, 33mto 169 1  |-  -.  (
( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4154   Oncon0 4523   suc csuc 4525   dom cdm 4819   ` cfv 5395    ~~ cen 7043    ~<_ cdom 7044    ~< csdm 7045   cardccrd 7756   alephcale 7757
This theorem is referenced by:  alephsucdom  7894  alephsucpw2  7926  alephgch  8487  winalim2  8505  aleph1re  12772
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-oi 7413  df-har 7460  df-card 7760  df-aleph 7761
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