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Theorem alephnbtwn2 8357
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 8244 . . 3  |-  ( card `  ( card `  B
) )  =  (
card `  B )
2 alephnbtwn 8356 . . 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 8354 . . . . . . . 8  |-  ( aleph ` 
suc  A )  e.  On
5 sdomdom 7450 . . . . . . . 8  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  ~<_  ( aleph ` 
suc  A ) )
6 ondomen 8322 . . . . . . . 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 8238 . . . . . . 7  |-  ( B  e.  dom  card  ->  (
card `  B )  ~~  B )
97, 8syl 16 . . . . . 6  |-  ( B 
~<  ( aleph `  suc  A )  ->  ( card `  B
)  ~~  B )
109ensymd 7473 . . . . 5  |-  ( B 
~<  ( aleph `  suc  A )  ->  B  ~~  ( card `  B ) )
11 sdomentr 7558 . . . . 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 8354 . . . . . 6  |-  ( aleph `  A )  e.  On
14 cardon 8229 . . . . . . 7  |-  ( card `  B )  e.  On
15 onenon 8234 . . . . . . 7  |-  ( (
card `  B )  e.  On  ->  ( card `  B )  e.  dom  card )
1614, 15ax-mp 5 . . . . . 6  |-  ( card `  B )  e.  dom  card
17 cardsdomel 8259 . . . . . 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 2532 . . . . 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 7560 . . . . . 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 8234 . . . . . . 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 8259 . . . . . 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 8355 . . . . . 6  |-  ( card `  ( aleph `  suc  A ) )  =  ( aleph ` 
suc  A )
3029eleq2i 2532 . . . . 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 1370    e. wcel 1758   class class class wbr 4403   Oncon0 4830   suc csuc 4832   dom cdm 4951   ` cfv 5529    ~~ cen 7420    ~<_ cdom 7421    ~< csdm 7422   cardccrd 8220   alephcale 8221
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-oi 7839  df-har 7888  df-card 8224  df-aleph 8225
This theorem is referenced by:  alephsucdom  8364  alephsucpw2  8396  alephgch  8956  winalim2  8978  aleph1re  13649
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