MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephnbtwn Structured version   Unicode version

Theorem alephnbtwn 8469
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 8467 . . . . . . . 8  |-  ( aleph `  A )  e.  On
2 id 22 . . . . . . . . . 10  |-  ( (
card `  B )  =  B  ->  ( card `  B )  =  B )
3 cardon 8342 . . . . . . . . . 10  |-  ( card `  B )  e.  On
42, 3syl6eqelr 2554 . . . . . . . . 9  |-  ( (
card `  B )  =  B  ->  B  e.  On )
5 onenon 8347 . . . . . . . . 9  |-  ( B  e.  On  ->  B  e.  dom  card )
64, 5syl 16 . . . . . . . 8  |-  ( (
card `  B )  =  B  ->  B  e. 
dom  card )
7 cardsdomel 8372 . . . . . . . 8  |-  ( ( ( aleph `  A )  e.  On  /\  B  e. 
dom  card )  ->  (
( aleph `  A )  ~<  B  <->  ( aleph `  A
)  e.  ( card `  B ) ) )
81, 6, 7sylancr 663 . . . . . . 7  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  ~<  B 
<->  ( aleph `  A )  e.  ( card `  B
) ) )
9 eleq2 2530 . . . . . . 7  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  e.  ( card `  B
)  <->  ( aleph `  A
)  e.  B ) )
108, 9bitrd 253 . . . . . 6  |-  ( (
card `  B )  =  B  ->  ( (
aleph `  A )  ~<  B 
<->  ( aleph `  A )  e.  B ) )
1110adantl 466 . . . . 5  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  ~<  B  <->  ( aleph `  A
)  e.  B ) )
12 alephsuc 8466 . . . . . . . . . . 11  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
13 onenon 8347 . . . . . . . . . . . 12  |-  ( (
aleph `  A )  e.  On  ->  ( aleph `  A )  e.  dom  card )
14 harval2 8395 . . . . . . . . . . . 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 2514 . . . . . . . . . 10  |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } )
1716eleq2d 2527 . . . . . . . . 9  |-  ( A  e.  On  ->  ( B  e.  ( aleph ` 
suc  A )  <->  B  e.  |^|
{ x  e.  On  |  ( aleph `  A
)  ~<  x } ) )
1817biimpd 207 . . . . . . . 8  |-  ( A  e.  On  ->  ( B  e.  ( aleph ` 
suc  A )  ->  B  e.  |^| { x  e.  On  |  ( aleph `  A )  ~<  x } ) )
19 breq2 4460 . . . . . . . . 9  |-  ( x  =  B  ->  (
( aleph `  A )  ~<  x  <->  ( aleph `  A
)  ~<  B ) )
2019onnminsb 6638 . . . . . . . 8  |-  ( B  e.  On  ->  ( B  e.  |^| { x  e.  On  |  ( aleph `  A )  ~<  x }  ->  -.  ( aleph `  A )  ~<  B ) )
2118, 20sylan9 657 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  e.  (
aleph `  suc  A )  ->  -.  ( aleph `  A )  ~<  B ) )
2221con2d 115 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  ~<  B  ->  -.  B  e.  ( aleph ` 
suc  A ) ) )
234, 22sylan2 474 . . . . 5  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  ~<  B  ->  -.  B  e.  ( aleph `  suc  A ) ) )
2411, 23sylbird 235 . . . 4  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  (
( aleph `  A )  e.  B  ->  -.  B  e.  ( aleph `  suc  A ) ) )
25 imnan 422 . . . 4  |-  ( ( ( aleph `  A )  e.  B  ->  -.  B  e.  ( aleph `  suc  A ) )  <->  -.  ( ( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) )
2624, 25sylib 196 . . 3  |-  ( ( A  e.  On  /\  ( card `  B )  =  B )  ->  -.  ( ( aleph `  A
)  e.  B  /\  B  e.  ( aleph ` 
suc  A ) ) )
2726ex 434 . 2  |-  ( A  e.  On  ->  (
( card `  B )  =  B  ->  -.  (
( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) ) ) )
28 n0i 3798 . . . . . . 7  |-  ( B  e.  ( aleph `  suc  A )  ->  -.  ( aleph `  suc  A )  =  (/) )
29 alephfnon 8463 . . . . . . . . . 10  |-  aleph  Fn  On
30 fndm 5686 . . . . . . . . . 10  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
3129, 30ax-mp 5 . . . . . . . . 9  |-  dom  aleph  =  On
3231eleq2i 2535 . . . . . . . 8  |-  ( suc 
A  e.  dom  aleph  <->  suc  A  e.  On )
33 ndmfv 5896 . . . . . . . 8  |-  ( -. 
suc  A  e.  dom  aleph  ->  ( aleph `  suc  A )  =  (/) )
3432, 33sylnbir 307 . . . . . . 7  |-  ( -. 
suc  A  e.  On  ->  ( aleph `  suc  A )  =  (/) )
3528, 34nsyl2 127 . . . . . 6  |-  ( B  e.  ( aleph `  suc  A )  ->  suc  A  e.  On )
36 sucelon 6651 . . . . . 6  |-  ( A  e.  On  <->  suc  A  e.  On )
3735, 36sylibr 212 . . . . 5  |-  ( B  e.  ( aleph `  suc  A )  ->  A  e.  On )
3837adantl 466 . . . 4  |-  ( ( ( aleph `  A )  e.  B  /\  B  e.  ( aleph `  suc  A ) )  ->  A  e.  On )
3938con3i 135 . . 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 164 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 184    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811   (/)c0 3793   |^|cint 4288   class class class wbr 4456   Oncon0 4887   suc csuc 4889   dom cdm 5008    Fn wfn 5589   ` cfv 5594    ~< csdm 7534  harchar 8000   cardccrd 8333   alephcale 8334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-oi 7953  df-har 8002  df-card 8337  df-aleph 8338
This theorem is referenced by:  alephnbtwn2  8470
  Copyright terms: Public domain W3C validator