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Theorem alephnbtwn 8520
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 alephon 8518 . . . . . . . 8  |-  ( aleph `  A )  e.  On
2 id 22 . . . . . . . . . 10  |-  ( (
card `  B )  =  B  ->  ( card `  B )  =  B )
3 cardon 8396 . . . . . . . . . 10  |-  ( card `  B )  e.  On
42, 3syl6eqelr 2558 . . . . . . . . 9  |-  ( (
card `  B )  =  B  ->  B  e.  On )
5 onenon 8401 . . . . . . . . 9  |-  ( B  e.  On  ->  B  e.  dom  card )
64, 5syl 17 . . . . . . . 8  |-  ( (
card `  B )  =  B  ->  B  e. 
dom  card )
7 cardsdomel 8426 . . . . . . . 8  |-  ( ( ( aleph `  A )  e.  On  /\  B  e. 
dom  card )  ->  (
( aleph `  A )  ~<  B  <->  ( aleph `  A
)  e.  ( card `  B ) ) )
81, 6, 7sylancr 676 . . . . . . 7  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  ~<  B 
<->  ( aleph `  A )  e.  ( card `  B
) ) )
9 eleq2 2538 . . . . . . 7  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  e.  ( card `  B
)  <->  ( aleph `  A
)  e.  B ) )
108, 9bitrd 261 . . . . . 6  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  ~<  B 
<->  ( aleph `  A )  e.  B ) )
1110adantl 473 . . . . 5  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  ~<  B  <->  ( aleph `  A
)  e.  B ) )
12 alephsuc 8517 . . . . . . . . . . 11  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
13 onenon 8401 . . . . . . . . . . . 12  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
14 harval2 8449 . . . . . . . . . . . 12  |-  ( (
aleph `  A )  e. 
dom  card  ->  (har `  ( aleph `  A ) )  =  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } )
151, 13, 14mp2b 10 . . . . . . . . . . 11  |-  (har `  ( aleph `  A )
)  =  |^| { x  e.  On  |  ( aleph `  A )  ~<  x }
1612, 15syl6eq 2521 . . . . . . . . . 10  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } )
1716eleq2d 2534 . . . . . . . . 9  |-  ( A  e.  On  ->  ( B  e.  ( aleph ` 
suc  A )  <->  B  e.  |^|
{ x  e.  On  |  ( aleph `  A
)  ~<  x } ) )
1817biimpd 212 . . . . . . . 8  |-  ( A  e.  On  ->  ( B  e.  ( aleph ` 
suc  A )  ->  B  e.  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } ) )
19 breq2 4399 . . . . . . . . 9  |-  ( x  =  B  ->  (
( aleph `  A )  ~<  x  <->  ( aleph `  A
)  ~<  B ) )
2019onnminsb 6650 . . . . . . . 8  |-  ( B  e.  On  ->  ( B  e.  |^| { x  e.  On  |  ( aleph `  A )  ~<  x }  ->  -.  ( aleph `  A )  ~<  B ) )
2118, 20sylan9 669 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  (
aleph `  suc  A )  ->  -.  ( aleph `  A )  ~<  B ) )
2221con2d 119 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<  B  ->  -.  B  e.  ( aleph ` 
suc  A ) ) )
234, 22sylan2 482 . . . . 5  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  ~<  B  ->  -.  B  e.  ( aleph `  suc  A ) ) )
2411, 23sylbird 243 . . . 4  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  e.  B  ->  -.  B  e.  ( aleph `  suc  A ) ) )
25 imnan 429 . . . 4  |-  ( ( ( aleph `  A )  e.  B  ->  -.  B  e.  ( aleph `  suc  A ) )  <->  -.  ( ( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) )
2624, 25sylib 201 . . 3  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  -.  ( ( aleph `  A
)  e.  B  /\  B  e.  ( aleph ` 
suc  A ) ) )
2726ex 441 . 2  |-  ( A  e.  On  ->  (
( card `  B )  =  B  ->  -.  (
( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) ) )
28 n0i 3727 . . . . . . 7  |-  ( B  e.  ( aleph `  suc  A )  ->  -.  ( aleph `  suc  A )  =  (/) )
29 alephfnon 8514 . . . . . . . . . 10  |-  aleph  Fn  On
30 fndm 5685 . . . . . . . . . 10  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
3129, 30ax-mp 5 . . . . . . . . 9  |-  dom  aleph  =  On
3231eleq2i 2541 . . . . . . . 8  |-  ( suc 
A  e.  dom  aleph  <->  suc  A  e.  On )
33 ndmfv 5903 . . . . . . . 8  |-  ( -. 
suc  A  e.  dom  aleph  ->  ( aleph `  suc  A )  =  (/) )
3432, 33sylnbir 314 . . . . . . 7  |-  ( -. 
suc  A  e.  On  ->  ( aleph `  suc  A )  =  (/) )
3528, 34nsyl2 132 . . . . . 6  |-  ( B  e.  ( aleph `  suc  A )  ->  suc  A  e.  On )
36 sucelon 6663 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
3735, 36sylibr 217 . . . . 5  |-  ( B  e.  ( aleph `  suc  A )  ->  A  e.  On )
3837adantl 473 . . . 4  |-  ( ( ( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) )  ->  A  e.  On )
3938con3i 142 . . 3  |-  ( -.  A  e.  On  ->  -.  ( ( aleph `  A
)  e.  B  /\  B  e.  ( aleph ` 
suc  A ) ) )
4039a1d 25 . 2  |-  ( -.  A  e.  On  ->  ( ( card `  B
)  =  B  ->  -.  ( ( aleph `  A
)  e.  B  /\  B  e.  ( aleph ` 
suc  A ) ) ) )
4127, 40pm2.61i 169 1  |-  ( (
card `  B )  =  B  ->  -.  (
( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   {crab 2760   (/)c0 3722   |^|cint 4226   class class class wbr 4395   dom cdm 4839   Oncon0 5430   suc csuc 5432    Fn wfn 5584   ` cfv 5589    ~< csdm 7586  harchar 8089   cardccrd 8387   alephcale 8388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-oi 8043  df-har 8091  df-card 8391  df-aleph 8392
This theorem is referenced by:  alephnbtwn2  8521
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