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Theorem alephnbtwn2 8451
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 8338 . . 3  |-  ( card `  ( card `  B
) )  =  (
card `  B )
2 alephnbtwn 8450 . . 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 8448 . . . . . . . 8  |-  ( aleph ` 
suc  A )  e.  On
5 sdomdom 7541 . . . . . . . 8  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  ~<_  ( aleph ` 
suc  A ) )
6 ondomen 8416 . . . . . . . 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 8332 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
97, 8syl 16 . . . . . 6  |-  ( B 
~<  ( aleph `  suc  A )  ->  ( card `  B
)  ~~  B )
109ensymd 7564 . . . . 5  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  ~~  ( card `  B ) )
11 sdomentr 7649 . . . . 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 8448 . . . . . 6  |-  ( aleph `  A )  e.  On
14 cardon 8323 . . . . . . 7  |-  ( card `  B )  e.  On
15 onenon 8328 . . . . . . 7  |-  ( (
card `  B )  e.  On  ->  ( card `  B )  e.  dom  card )
1614, 15ax-mp 5 . . . . . 6  |-  ( card `  B )  e.  dom  card
17 cardsdomel 8353 . . . . . 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 2519 . . . . 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 7651 . . . . . 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 8328 . . . . . . 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 8353 . . . . . 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 8449 . . . . . 6  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
3029eleq2i 2519 . . . . 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 1381    e. wcel 1802   class class class wbr 4433   Oncon0 4864   suc csuc 4866   dom cdm 4985   ` cfv 5574    ~~ cen 7511    ~<_ cdom 7512    ~< csdm 7513   cardccrd 8314   alephcale 8315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-om 6682  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-oi 7933  df-har 7982  df-card 8318  df-aleph 8319
This theorem is referenced by:  alephsucdom  8458  alephsucpw2  8490  alephgch  9050  winalim2  9072  aleph1re  13850
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