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Theorem winalim2 9072
Description: A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)
Assertion
Ref Expression
winalim2  |-  ( ( A  e.  InaccW  /\  A  =/=  om )  ->  E. x ( ( aleph `  x )  =  A  /\  Lim  x ) )
Distinct variable group:    x, A

Proof of Theorem winalim2
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 winacard 9068 . . . 4  |-  ( A  e.  InaccW  ->  ( card `  A )  =  A )
2 winainf 9070 . . . . 5  |-  ( A  e.  InaccW  ->  om  C_  A
)
3 cardalephex 8472 . . . . 5  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
42, 3syl 17 . . . 4  |-  ( A  e.  InaccW  ->  (
( card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
51, 4mpbid 213 . . 3  |-  ( A  e.  InaccW  ->  E. x  e.  On  A  =  (
aleph `  x ) )
65adantr 466 . 2  |-  ( ( A  e.  InaccW  /\  A  =/=  om )  ->  E. x  e.  On  A  =  ( aleph `  x ) )
7 df-rex 2720 . . 3  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  <->  E. x ( x  e.  On  /\  A  =  ( aleph `  x
) ) )
8 simprr 764 . . . . . . 7  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  A  =  ( aleph `  x
) )
98eqcomd 2434 . . . . . 6  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  ( aleph `  x )  =  A )
10 simprl 762 . . . . . . . 8  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  x  e.  On )
11 onzsl 6631 . . . . . . . 8  |-  ( x  e.  On  <->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y  \/  ( x  e.  _V  /\  Lim  x
) ) )
1210, 11sylib 199 . . . . . . 7  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
x  =  (/)  \/  E. y  e.  On  x  =  suc  y  \/  (
x  e.  _V  /\  Lim  x ) ) )
13 simplr 760 . . . . . . . . . 10  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  A  =/=  om )
14 fveq2 5825 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
15 aleph0 8448 . . . . . . . . . . . . . 14  |-  ( aleph `  (/) )  =  om
1614, 15syl6eq 2478 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( aleph `  x )  =  om )
17 eqtr 2447 . . . . . . . . . . . . 13  |-  ( ( A  =  ( aleph `  x )  /\  ( aleph `  x )  =  om )  ->  A  =  om )
1816, 17sylan2 476 . . . . . . . . . . . 12  |-  ( ( A  =  ( aleph `  x )  /\  x  =  (/) )  ->  A  =  om )
1918ex 435 . . . . . . . . . . 11  |-  ( A  =  ( aleph `  x
)  ->  ( x  =  (/)  ->  A  =  om ) )
2019necon3ad 2614 . . . . . . . . . 10  |-  ( A  =  ( aleph `  x
)  ->  ( A  =/=  om  ->  -.  x  =  (/) ) )
218, 13, 20sylc 62 . . . . . . . . 9  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  -.  x  =  (/) )
2221pm2.21d 109 . . . . . . . 8  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
x  =  (/)  ->  Lim  x ) )
23 suceloni 6598 . . . . . . . . . . . . . . . 16  |-  ( y  e.  On  ->  suc  y  e.  On )
24 vex 3025 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
2524sucid 5464 . . . . . . . . . . . . . . . 16  |-  y  e. 
suc  y
26 alephord2i 8459 . . . . . . . . . . . . . . . 16  |-  ( suc  y  e.  On  ->  ( y  e.  suc  y  ->  ( aleph `  y )  e.  ( aleph `  suc  y ) ) )
2723, 25, 26mpisyl 21 . . . . . . . . . . . . . . 15  |-  ( y  e.  On  ->  ( aleph `  y )  e.  ( aleph `  suc  y ) )
2827ad2antrl 732 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( aleph `  y
)  e.  ( aleph ` 
suc  y ) )
29 simplrr 769 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  A  =  (
aleph `  x ) )
30 fveq2 5825 . . . . . . . . . . . . . . . 16  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
3130ad2antll 733 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( aleph `  x
)  =  ( aleph ` 
suc  y ) )
3229, 31eqtrd 2462 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  A  =  (
aleph `  suc  y ) )
3328, 32eleqtrrd 2509 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( aleph `  y
)  e.  A )
34 elwina 9062 . . . . . . . . . . . . . . 15  |-  ( A  e.  InaccW  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. z  e.  A  E. w  e.  A  z  ~<  w
) )
3534simp3bi 1022 . . . . . . . . . . . . . 14  |-  ( A  e.  InaccW  ->  A. z  e.  A  E. w  e.  A  z  ~<  w )
3635ad3antrrr 734 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  A. z  e.  A  E. w  e.  A  z  ~<  w )
37 breq1 4369 . . . . . . . . . . . . . . 15  |-  ( z  =  ( aleph `  y
)  ->  ( z  ~<  w  <->  ( aleph `  y
)  ~<  w ) )
3837rexbidv 2878 . . . . . . . . . . . . . 14  |-  ( z  =  ( aleph `  y
)  ->  ( E. w  e.  A  z  ~<  w  <->  E. w  e.  A  ( aleph `  y )  ~<  w ) )
3938rspcva 3123 . . . . . . . . . . . . 13  |-  ( ( ( aleph `  y )  e.  A  /\  A. z  e.  A  E. w  e.  A  z  ~<  w )  ->  E. w  e.  A  ( aleph `  y )  ~<  w
)
4033, 36, 39syl2anc 665 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  E. w  e.  A  ( aleph `  y )  ~<  w )
4140expr 618 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  y  e.  On )  ->  ( x  =  suc  y  ->  E. w  e.  A  ( aleph `  y )  ~<  w
) )
42 iscard 8361 . . . . . . . . . . . . . . . . . . 19  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. w  e.  A  w  ~<  A ) )
4342simprbi 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
card `  A )  =  A  ->  A. w  e.  A  w  ~<  A )
44 rsp 2731 . . . . . . . . . . . . . . . . . 18  |-  ( A. w  e.  A  w  ~<  A  ->  ( w  e.  A  ->  w  ~<  A ) )
451, 43, 443syl 18 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  InaccW  ->  (
w  e.  A  ->  w  ~<  A ) )
4645ad3antrrr 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( w  e.  A  ->  w  ~<  A ) )
4732breq2d 4378 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( w  ~<  A  <-> 
w  ~<  ( aleph `  suc  y ) ) )
4846, 47sylibd 217 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( w  e.  A  ->  w  ~<  (
aleph `  suc  y ) ) )
49 alephnbtwn2 8454 . . . . . . . . . . . . . . . 16  |-  -.  (
( aleph `  y )  ~<  w  /\  w  ~<  (
aleph `  suc  y ) )
50 pm3.21 449 . . . . . . . . . . . . . . . 16  |-  ( w 
~<  ( aleph `  suc  y )  ->  ( ( aleph `  y )  ~<  w  ->  ( ( aleph `  y
)  ~<  w  /\  w  ~<  ( aleph `  suc  y ) ) ) )
5149, 50mtoi 181 . . . . . . . . . . . . . . 15  |-  ( w 
~<  ( aleph `  suc  y )  ->  -.  ( aleph `  y )  ~<  w
)
5248, 51syl6 34 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( w  e.  A  ->  -.  ( aleph `  y )  ~<  w ) )
5352imp 430 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  /\  w  e.  A
)  ->  -.  ( aleph `  y )  ~<  w )
5453nrexdv 2820 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  -.  E. w  e.  A  ( aleph `  y )  ~<  w
)
5554expr 618 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  y  e.  On )  ->  ( x  =  suc  y  ->  -.  E. w  e.  A  (
aleph `  y )  ~<  w ) )
5641, 55pm2.65d 178 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  y  e.  On )  ->  -.  x  =  suc  y )
5756nrexdv 2820 . . . . . . . . 9  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  -.  E. y  e.  On  x  =  suc  y )
5857pm2.21d 109 . . . . . . . 8  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  ( E. y  e.  On  x  =  suc  y  ->  Lim  x ) )
59 simpr 462 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  Lim  x )  ->  Lim  x )
6059a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
( x  e.  _V  /\ 
Lim  x )  ->  Lim  x ) )
6122, 58, 603jaod 1328 . . . . . . 7  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y  \/  ( x  e.  _V  /\ 
Lim  x ) )  ->  Lim  x )
)
6212, 61mpd 15 . . . . . 6  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  Lim  x )
639, 62jca 534 . . . . 5  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
( aleph `  x )  =  A  /\  Lim  x
) )
6463ex 435 . . . 4  |-  ( ( A  e.  InaccW  /\  A  =/=  om )  -> 
( ( x  e.  On  /\  A  =  ( aleph `  x )
)  ->  ( ( aleph `  x )  =  A  /\  Lim  x
) ) )
6564eximdv 1758 . . 3  |-  ( ( A  e.  InaccW  /\  A  =/=  om )  -> 
( E. x ( x  e.  On  /\  A  =  ( aleph `  x ) )  ->  E. x ( ( aleph `  x )  =  A  /\  Lim  x ) ) )
667, 65syl5bi 220 . 2  |-  ( ( A  e.  InaccW  /\  A  =/=  om )  -> 
( E. x  e.  On  A  =  (
aleph `  x )  ->  E. x ( ( aleph `  x )  =  A  /\  Lim  x ) ) )
676, 66mpd 15 1  |-  ( ( A  e.  InaccW  /\  A  =/=  om )  ->  E. x ( ( aleph `  x )  =  A  /\  Lim  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    \/ w3o 981    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715   _Vcvv 3022    C_ wss 3379   (/)c0 3704   class class class wbr 4366   Oncon0 5385   Lim wlim 5386   suc csuc 5387   ` cfv 5544   omcom 6650    ~< csdm 7523   cardccrd 8321   alephcale 8322   cfccf 8323   InaccWcwina 9058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-oi 7978  df-har 8026  df-card 8325  df-aleph 8326  df-cf 8327  df-wina 9060
This theorem is referenced by:  winafp  9073
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