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Theorem alephnbtwn2 8465
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 8352 . . 3  |-  ( card `  ( card `  B
) )  =  (
card `  B )
2 alephnbtwn 8464 . . 3  |-  ( (
card `  ( card `  B ) )  =  ( card `  B
)  ->  -.  (
( aleph `  A )  e.  ( card `  B
)  /\  ( card `  B )  e.  (
aleph `  suc  A ) ) )
31, 2ax-mp 5 . 2  |-  -.  (
( aleph `  A )  e.  ( card `  B
)  /\  ( card `  B )  e.  (
aleph `  suc  A ) )
4 alephon 8462 . . . . . . . 8  |-  ( aleph ` 
suc  A )  e.  On
5 sdomdom 7555 . . . . . . . 8  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  ~<_  ( aleph ` 
suc  A ) )
6 ondomen 8430 . . . . . . . 8  |-  ( ( ( aleph `  suc  A )  e.  On  /\  B  ~<_  ( aleph `  suc  A ) )  ->  B  e.  dom  card )
74, 5, 6sylancr 663 . . . . . . 7  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  e.  dom  card )
8 cardid2 8346 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
97, 8syl 16 . . . . . 6  |-  ( B 
~<  ( aleph `  suc  A )  ->  ( card `  B
)  ~~  B )
109ensymd 7578 . . . . 5  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  ~~  ( card `  B ) )
11 sdomentr 7663 . . . . 5  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~~  ( card `  B )
)  ->  ( aleph `  A )  ~<  ( card `  B ) )
1210, 11sylan2 474 . . . 4  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( aleph `  A )  ~<  ( card `  B ) )
13 alephon 8462 . . . . . 6  |-  ( aleph `  A )  e.  On
14 cardon 8337 . . . . . . 7  |-  ( card `  B )  e.  On
15 onenon 8342 . . . . . . 7  |-  ( (
card `  B )  e.  On  ->  ( card `  B )  e.  dom  card )
1614, 15ax-mp 5 . . . . . 6  |-  ( card `  B )  e.  dom  card
17 cardsdomel 8367 . . . . . 6  |-  ( ( ( aleph `  A )  e.  On  /\  ( card `  B )  e.  dom  card )  ->  ( ( aleph `  A )  ~< 
( card `  B )  <->  (
aleph `  A )  e.  ( card `  ( card `  B ) ) ) )
1813, 16, 17mp2an 672 . . . . 5  |-  ( (
aleph `  A )  ~< 
( card `  B )  <->  (
aleph `  A )  e.  ( card `  ( card `  B ) ) )
191eleq2i 2545 . . . . 5  |-  ( (
aleph `  A )  e.  ( card `  ( card `  B ) )  <-> 
( aleph `  A )  e.  ( card `  B
) )
2018, 19bitri 249 . . . 4  |-  ( (
aleph `  A )  ~< 
( card `  B )  <->  (
aleph `  A )  e.  ( card `  B
) )
2112, 20sylib 196 . . 3  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( aleph `  A )  e.  (
card `  B )
)
22 ensdomtr 7665 . . . . . 6  |-  ( ( ( card `  B
)  ~~  B  /\  B  ~<  ( aleph `  suc  A ) )  ->  ( card `  B )  ~< 
( aleph `  suc  A ) )
239, 22mpancom 669 . . . . 5  |-  ( B 
~<  ( aleph `  suc  A )  ->  ( card `  B
)  ~<  ( aleph `  suc  A ) )
2423adantl 466 . . . 4  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( card `  B )  ~<  ( aleph `  suc  A ) )
25 onenon 8342 . . . . . . 7  |-  ( (
aleph `  suc  A )  e.  On  ->  ( aleph `  suc  A )  e.  dom  card )
264, 25ax-mp 5 . . . . . 6  |-  ( aleph ` 
suc  A )  e. 
dom  card
27 cardsdomel 8367 . . . . . 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 672 . . . . 5  |-  ( (
card `  B )  ~<  ( aleph `  suc  A )  <-> 
( card `  B )  e.  ( card `  ( aleph `  suc  A ) ) )
29 alephcard 8463 . . . . . 6  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
3029eleq2i 2545 . . . . 5  |-  ( (
card `  B )  e.  ( card `  ( aleph `  suc  A ) )  <->  ( card `  B
)  e.  ( aleph ` 
suc  A ) )
3128, 30bitri 249 . . . 4  |-  ( (
card `  B )  ~<  ( aleph `  suc  A )  <-> 
( card `  B )  e.  ( aleph `  suc  A ) )
3224, 31sylib 196 . . 3  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( card `  B )  e.  (
aleph `  suc  A ) )
3321, 32jca 532 . 2  |-  ( ( ( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )  ->  ( ( aleph `  A )  e.  ( card `  B
)  /\  ( card `  B )  e.  (
aleph `  suc  A ) ) )
343, 33mto 176 1  |-  -.  (
( aleph `  A )  ~<  B  /\  B  ~<  (
aleph `  suc  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4453   Oncon0 4884   suc csuc 4886   dom cdm 5005   ` cfv 5594    ~~ cen 7525    ~<_ cdom 7526    ~< csdm 7527   cardccrd 8328   alephcale 8329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-oi 7947  df-har 7996  df-card 8332  df-aleph 8333
This theorem is referenced by:  alephsucdom  8472  alephsucpw2  8504  alephgch  9064  winalim2  9086  aleph1re  13856
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