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Theorem alephnbtwn 7941
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 7939 . . . . . . . 8  |-  ( aleph `  A )  e.  On
2 id 20 . . . . . . . . . 10  |-  ( (
card `  B )  =  B  ->  ( card `  B )  =  B )
3 cardon 7820 . . . . . . . . . 10  |-  ( card `  B )  e.  On
42, 3syl6eqelr 2524 . . . . . . . . 9  |-  ( (
card `  B )  =  B  ->  B  e.  On )
5 onenon 7825 . . . . . . . . 9  |-  ( B  e.  On  ->  B  e.  dom  card )
64, 5syl 16 . . . . . . . 8  |-  ( (
card `  B )  =  B  ->  B  e. 
dom  card )
7 cardsdomel 7850 . . . . . . . 8  |-  ( ( ( aleph `  A )  e.  On  /\  B  e. 
dom  card )  ->  (
( aleph `  A )  ~<  B  <->  ( aleph `  A
)  e.  ( card `  B ) ) )
81, 6, 7sylancr 645 . . . . . . 7  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  ~<  B 
<->  ( aleph `  A )  e.  ( card `  B
) ) )
9 eleq2 2496 . . . . . . 7  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  e.  ( card `  B
)  <->  ( aleph `  A
)  e.  B ) )
108, 9bitrd 245 . . . . . 6  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  ~<  B 
<->  ( aleph `  A )  e.  B ) )
1110adantl 453 . . . . 5  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  ~<  B  <->  ( aleph `  A
)  e.  B ) )
12 alephsuc 7938 . . . . . . . . . . 11  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
13 onenon 7825 . . . . . . . . . . . 12  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
14 harval2 7873 . . . . . . . . . . . 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 2483 . . . . . . . . . 10  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } )
1716eleq2d 2502 . . . . . . . . 9  |-  ( A  e.  On  ->  ( B  e.  ( aleph ` 
suc  A )  <->  B  e.  |^|
{ x  e.  On  |  ( aleph `  A
)  ~<  x } ) )
1817biimpd 199 . . . . . . . 8  |-  ( A  e.  On  ->  ( B  e.  ( aleph ` 
suc  A )  ->  B  e.  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } ) )
19 breq2 4208 . . . . . . . . 9  |-  ( x  =  B  ->  (
( aleph `  A )  ~<  x  <->  ( aleph `  A
)  ~<  B ) )
2019onnminsb 4775 . . . . . . . 8  |-  ( B  e.  On  ->  ( B  e.  |^| { x  e.  On  |  ( aleph `  A )  ~<  x }  ->  -.  ( aleph `  A )  ~<  B ) )
2118, 20sylan9 639 . . . . . . 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 461 . . . . 5  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  ~<  B  ->  -.  B  e.  ( aleph `  suc  A ) ) )
2411, 23sylbird 227 . . . 4  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  e.  B  ->  -.  B  e.  ( aleph `  suc  A ) ) )
25 imnan 412 . . . 4  |-  ( ( ( aleph `  A )  e.  B  ->  -.  B  e.  ( aleph `  suc  A ) )  <->  -.  ( ( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) )
2624, 25sylib 189 . . 3  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  -.  ( ( aleph `  A
)  e.  B  /\  B  e.  ( aleph ` 
suc  A ) ) )
2726ex 424 . 2  |-  ( A  e.  On  ->  (
( card `  B )  =  B  ->  -.  (
( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) ) )
28 n0i 3625 . . . . . . 7  |-  ( B  e.  ( aleph `  suc  A )  ->  -.  ( aleph `  suc  A )  =  (/) )
29 alephfnon 7935 . . . . . . . . . 10  |-  aleph  Fn  On
30 fndm 5535 . . . . . . . . . 10  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
3129, 30ax-mp 8 . . . . . . . . 9  |-  dom  aleph  =  On
3231eleq2i 2499 . . . . . . . 8  |-  ( suc 
A  e.  dom  aleph  <->  suc  A  e.  On )
33 ndmfv 5746 . . . . . . . 8  |-  ( -. 
suc  A  e.  dom  aleph  ->  ( aleph `  suc  A )  =  (/) )
3432, 33sylnbir 299 . . . . . . 7  |-  ( -. 
suc  A  e.  On  ->  ( aleph `  suc  A )  =  (/) )
3528, 34nsyl2 121 . . . . . 6  |-  ( B  e.  ( aleph `  suc  A )  ->  suc  A  e.  On )
36 sucelon 4788 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
3735, 36sylibr 204 . . . . 5  |-  ( B  e.  ( aleph `  suc  A )  ->  A  e.  On )
3837adantl 453 . . . 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 23 . 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 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   (/)c0 3620   |^|cint 4042   class class class wbr 4204   Oncon0 4573   suc csuc 4575   dom cdm 4869    Fn wfn 5440   ` cfv 5445    ~< csdm 7099  harchar 7513   cardccrd 7811   alephcale 7812
This theorem is referenced by:  alephnbtwn2  7942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-oi 7468  df-har 7515  df-card 7815  df-aleph 7816
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