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Theorem alephnbtwn 8502
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 8500 . . . . . . . 8  |-  ( aleph `  A )  e.  On
2 id 22 . . . . . . . . . 10  |-  ( (
card `  B )  =  B  ->  ( card `  B )  =  B )
3 cardon 8378 . . . . . . . . . 10  |-  ( card `  B )  e.  On
42, 3syl6eqelr 2538 . . . . . . . . 9  |-  ( (
card `  B )  =  B  ->  B  e.  On )
5 onenon 8383 . . . . . . . . 9  |-  ( B  e.  On  ->  B  e.  dom  card )
64, 5syl 17 . . . . . . . 8  |-  ( (
card `  B )  =  B  ->  B  e. 
dom  card )
7 cardsdomel 8408 . . . . . . . 8  |-  ( ( ( aleph `  A )  e.  On  /\  B  e. 
dom  card )  ->  (
( aleph `  A )  ~<  B  <->  ( aleph `  A
)  e.  ( card `  B ) ) )
81, 6, 7sylancr 669 . . . . . . 7  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  ~<  B 
<->  ( aleph `  A )  e.  ( card `  B
) ) )
9 eleq2 2518 . . . . . . 7  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  e.  ( card `  B
)  <->  ( aleph `  A
)  e.  B ) )
108, 9bitrd 257 . . . . . 6  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  ~<  B 
<->  ( aleph `  A )  e.  B ) )
1110adantl 468 . . . . 5  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  ~<  B  <->  ( aleph `  A
)  e.  B ) )
12 alephsuc 8499 . . . . . . . . . . 11  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
13 onenon 8383 . . . . . . . . . . . 12  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
14 harval2 8431 . . . . . . . . . . . 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 2501 . . . . . . . . . 10  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } )
1716eleq2d 2514 . . . . . . . . 9  |-  ( A  e.  On  ->  ( B  e.  ( aleph ` 
suc  A )  <->  B  e.  |^|
{ x  e.  On  |  ( aleph `  A
)  ~<  x } ) )
1817biimpd 211 . . . . . . . 8  |-  ( A  e.  On  ->  ( B  e.  ( aleph ` 
suc  A )  ->  B  e.  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } ) )
19 breq2 4406 . . . . . . . . 9  |-  ( x  =  B  ->  (
( aleph `  A )  ~<  x  <->  ( aleph `  A
)  ~<  B ) )
2019onnminsb 6631 . . . . . . . 8  |-  ( B  e.  On  ->  ( B  e.  |^| { x  e.  On  |  ( aleph `  A )  ~<  x }  ->  -.  ( aleph `  A )  ~<  B ) )
2118, 20sylan9 663 . . . . . . 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 477 . . . . 5  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  ~<  B  ->  -.  B  e.  ( aleph `  suc  A ) ) )
2411, 23sylbird 239 . . . 4  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  e.  B  ->  -.  B  e.  ( aleph `  suc  A ) ) )
25 imnan 424 . . . 4  |-  ( ( ( aleph `  A )  e.  B  ->  -.  B  e.  ( aleph `  suc  A ) )  <->  -.  ( ( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) )
2624, 25sylib 200 . . 3  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  -.  ( ( aleph `  A
)  e.  B  /\  B  e.  ( aleph ` 
suc  A ) ) )
2726ex 436 . 2  |-  ( A  e.  On  ->  (
( card `  B )  =  B  ->  -.  (
( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) ) )
28 n0i 3736 . . . . . . 7  |-  ( B  e.  ( aleph `  suc  A )  ->  -.  ( aleph `  suc  A )  =  (/) )
29 alephfnon 8496 . . . . . . . . . 10  |-  aleph  Fn  On
30 fndm 5675 . . . . . . . . . 10  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
3129, 30ax-mp 5 . . . . . . . . 9  |-  dom  aleph  =  On
3231eleq2i 2521 . . . . . . . 8  |-  ( suc 
A  e.  dom  aleph  <->  suc  A  e.  On )
33 ndmfv 5889 . . . . . . . 8  |-  ( -. 
suc  A  e.  dom  aleph  ->  ( aleph `  suc  A )  =  (/) )
3432, 33sylnbir 309 . . . . . . 7  |-  ( -. 
suc  A  e.  On  ->  ( aleph `  suc  A )  =  (/) )
3528, 34nsyl2 131 . . . . . 6  |-  ( B  e.  ( aleph `  suc  A )  ->  suc  A  e.  On )
36 sucelon 6644 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
3735, 36sylibr 216 . . . . 5  |-  ( B  e.  ( aleph `  suc  A )  ->  A  e.  On )
3837adantl 468 . . . 4  |-  ( ( ( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) )  ->  A  e.  On )
3938con3i 141 . . 3  |-  ( -.  A  e.  On  ->  -.  ( ( aleph `  A
)  e.  B  /\  B  e.  ( aleph ` 
suc  A ) ) )
4039a1d 26 . 2  |-  ( -.  A  e.  On  ->  ( ( card `  B
)  =  B  ->  -.  ( ( aleph `  A
)  e.  B  /\  B  e.  ( aleph ` 
suc  A ) ) ) )
4127, 40pm2.61i 168 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   {crab 2741   (/)c0 3731   |^|cint 4234   class class class wbr 4402   dom cdm 4834   Oncon0 5423   suc csuc 5425    Fn wfn 5577   ` cfv 5582    ~< csdm 7568  harchar 8071   cardccrd 8369   alephcale 8370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-oi 8025  df-har 8073  df-card 8373  df-aleph 8374
This theorem is referenced by:  alephnbtwn2  8503
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