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Theorem alephnbtwn 7652
Description: No cardinal can be sandwiched between an aleph and its successor aleph. Theorem 67 of [Suppes] p. 229. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
alephnbtwn  |-  ( (
card `  B )  =  B  ->  -.  (
( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) )

Proof of Theorem alephnbtwn
StepHypRef Expression
1 alephon 7650 . . . . . . . 8  |-  ( aleph `  A )  e.  On
2 id 21 . . . . . . . . . 10  |-  ( (
card `  B )  =  B  ->  ( card `  B )  =  B )
3 cardon 7531 . . . . . . . . . 10  |-  ( card `  B )  e.  On
42, 3syl6eqelr 2345 . . . . . . . . 9  |-  ( (
card `  B )  =  B  ->  B  e.  On )
5 onenon 7536 . . . . . . . . 9  |-  ( B  e.  On  ->  B  e.  dom  card )
64, 5syl 17 . . . . . . . 8  |-  ( (
card `  B )  =  B  ->  B  e. 
dom  card )
7 cardsdomel 7561 . . . . . . . 8  |-  ( ( ( aleph `  A )  e.  On  /\  B  e. 
dom  card )  ->  (
( aleph `  A )  ~<  B  <->  ( aleph `  A
)  e.  ( card `  B ) ) )
81, 6, 7sylancr 647 . . . . . . 7  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  ~<  B 
<->  ( aleph `  A )  e.  ( card `  B
) ) )
9 eleq2 2317 . . . . . . 7  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  e.  ( card `  B
)  <->  ( aleph `  A
)  e.  B ) )
108, 9bitrd 246 . . . . . 6  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  ~<  B 
<->  ( aleph `  A )  e.  B ) )
1110adantl 454 . . . . 5  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  ~<  B  <->  ( aleph `  A
)  e.  B ) )
12 alephsuc 7649 . . . . . . . . . . 11  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
13 onenon 7536 . . . . . . . . . . . 12  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
14 harval2 7584 . . . . . . . . . . . 12  |-  ( (
aleph `  A )  e. 
dom  card  ->  (har `  ( aleph `  A ) )  =  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } )
151, 13, 14mp2b 11 . . . . . . . . . . 11  |-  (har `  ( aleph `  A )
)  =  |^| { x  e.  On  |  ( aleph `  A )  ~<  x }
1612, 15syl6eq 2304 . . . . . . . . . 10  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } )
1716eleq2d 2323 . . . . . . . . 9  |-  ( A  e.  On  ->  ( B  e.  ( aleph ` 
suc  A )  <->  B  e.  |^|
{ x  e.  On  |  ( aleph `  A
)  ~<  x } ) )
1817biimpd 200 . . . . . . . 8  |-  ( A  e.  On  ->  ( B  e.  ( aleph ` 
suc  A )  ->  B  e.  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } ) )
19 breq2 3987 . . . . . . . . 9  |-  ( x  =  B  ->  (
( aleph `  A )  ~<  x  <->  ( aleph `  A
)  ~<  B ) )
2019onnminsb 4553 . . . . . . . 8  |-  ( B  e.  On  ->  ( B  e.  |^| { x  e.  On  |  ( aleph `  A )  ~<  x }  ->  -.  ( aleph `  A )  ~<  B ) )
2118, 20sylan9 641 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  (
aleph `  suc  A )  ->  -.  ( aleph `  A )  ~<  B ) )
2221con2d 109 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<  B  ->  -.  B  e.  ( aleph ` 
suc  A ) ) )
234, 22sylan2 462 . . . . 5  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  ~<  B  ->  -.  B  e.  ( aleph `  suc  A ) ) )
2411, 23sylbird 228 . . . 4  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  e.  B  ->  -.  B  e.  ( aleph `  suc  A ) ) )
25 imnan 413 . . . 4  |-  ( ( ( aleph `  A )  e.  B  ->  -.  B  e.  ( aleph `  suc  A ) )  <->  -.  ( ( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) )
2624, 25sylib 190 . . 3  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  -.  ( ( aleph `  A
)  e.  B  /\  B  e.  ( aleph ` 
suc  A ) ) )
2726ex 425 . 2  |-  ( A  e.  On  ->  (
( card `  B )  =  B  ->  -.  (
( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) ) )
28 n0i 3421 . . . . . . 7  |-  ( B  e.  ( aleph `  suc  A )  ->  -.  ( aleph `  suc  A )  =  (/) )
29 alephfnon 7646 . . . . . . . . . 10  |-  aleph  Fn  On
30 fndm 5267 . . . . . . . . . 10  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
3129, 30ax-mp 10 . . . . . . . . 9  |-  dom  aleph  =  On
3231eleq2i 2320 . . . . . . . 8  |-  ( suc 
A  e.  dom  aleph  <->  suc  A  e.  On )
33 ndmfv 5472 . . . . . . . 8  |-  ( -. 
suc  A  e.  dom  aleph  ->  ( aleph `  suc  A )  =  (/) )
3432, 33sylnbir 300 . . . . . . 7  |-  ( -. 
suc  A  e.  On  ->  ( aleph `  suc  A )  =  (/) )
3528, 34nsyl2 121 . . . . . 6  |-  ( B  e.  ( aleph `  suc  A )  ->  suc  A  e.  On )
36 sucelon 4566 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
3735, 36sylibr 205 . . . . 5  |-  ( B  e.  ( aleph `  suc  A )  ->  A  e.  On )
3837adantl 454 . . . 4  |-  ( ( ( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) )  ->  A  e.  On )
3938con3i 129 . . 3  |-  ( -.  A  e.  On  ->  -.  ( ( aleph `  A
)  e.  B  /\  B  e.  ( aleph ` 
suc  A ) ) )
4039a1d 24 . 2  |-  ( -.  A  e.  On  ->  ( ( card `  B
)  =  B  ->  -.  ( ( aleph `  A
)  e.  B  /\  B  e.  ( aleph ` 
suc  A ) ) ) )
4127, 40pm2.61i 158 1  |-  ( (
card `  B )  =  B  ->  -.  (
( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   {crab 2520   (/)c0 3416   |^|cint 3822   class class class wbr 3983   Oncon0 4350   suc csuc 4352   dom cdm 4647    Fn wfn 4654   ` cfv 4659    ~< csdm 6816  harchar 7224   cardccrd 7522   alephcale 7523
This theorem is referenced by:  alephnbtwn2  7653
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-oi 7179  df-har 7226  df-card 7526  df-aleph 7527
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