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Theorem winalim2 8868
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 8864 . . . 4  |-  ( A  e.  InaccW  ->  ( card `  A )  =  A )
2 winainf 8866 . . . . 5  |-  ( A  e.  InaccW  ->  om  C_  A
)
3 cardalephex 8265 . . . . 5  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
42, 3syl 16 . . . 4  |-  ( A  e.  InaccW  ->  (
( card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
51, 4mpbid 210 . . 3  |-  ( A  e.  InaccW  ->  E. x  e.  On  A  =  (
aleph `  x ) )
65adantr 465 . 2  |-  ( ( A  e.  InaccW  /\  A  =/=  om )  ->  E. x  e.  On  A  =  ( aleph `  x ) )
7 df-rex 2726 . . 3  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  <->  E. x ( x  e.  On  /\  A  =  ( aleph `  x
) ) )
8 simprr 756 . . . . . . 7  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  A  =  ( aleph `  x
) )
98eqcomd 2448 . . . . . 6  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  ( aleph `  x )  =  A )
10 simprl 755 . . . . . . . 8  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  x  e.  On )
11 onzsl 6462 . . . . . . . 8  |-  ( x  e.  On  <->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y  \/  ( x  e.  _V  /\  Lim  x
) ) )
1210, 11sylib 196 . . . . . . 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 754 . . . . . . . . . 10  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  A  =/=  om )
14 fveq2 5696 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
15 aleph0 8241 . . . . . . . . . . . . . 14  |-  ( aleph `  (/) )  =  om
1614, 15syl6eq 2491 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( aleph `  x )  =  om )
17 eqtr 2460 . . . . . . . . . . . . 13  |-  ( ( A  =  ( aleph `  x )  /\  ( aleph `  x )  =  om )  ->  A  =  om )
1816, 17sylan2 474 . . . . . . . . . . . 12  |-  ( ( A  =  ( aleph `  x )  /\  x  =  (/) )  ->  A  =  om )
1918ex 434 . . . . . . . . . . 11  |-  ( A  =  ( aleph `  x
)  ->  ( x  =  (/)  ->  A  =  om ) )
2019necon3ad 2649 . . . . . . . . . 10  |-  ( A  =  ( aleph `  x
)  ->  ( A  =/=  om  ->  -.  x  =  (/) ) )
218, 13, 20sylc 60 . . . . . . . . 9  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  -.  x  =  (/) )
2221pm2.21d 106 . . . . . . . 8  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
x  =  (/)  ->  Lim  x ) )
23 suceloni 6429 . . . . . . . . . . . . . . . 16  |-  ( y  e.  On  ->  suc  y  e.  On )
24 vex 2980 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
2524sucid 4803 . . . . . . . . . . . . . . . 16  |-  y  e. 
suc  y
26 alephord2i 8252 . . . . . . . . . . . . . . . 16  |-  ( suc  y  e.  On  ->  ( y  e.  suc  y  ->  ( aleph `  y )  e.  ( aleph `  suc  y ) ) )
2723, 25, 26mpisyl 18 . . . . . . . . . . . . . . 15  |-  ( y  e.  On  ->  ( aleph `  y )  e.  ( aleph `  suc  y ) )
2827ad2antrl 727 . . . . . . . . . . . . . 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 760 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  A  =  (
aleph `  x ) )
30 fveq2 5696 . . . . . . . . . . . . . . . 16  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
3130ad2antll 728 . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  A  =  (
aleph `  suc  y ) )
3328, 32eleqtrrd 2520 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( aleph `  y
)  e.  A )
34 elwina 8858 . . . . . . . . . . . . . . 15  |-  ( A  e.  InaccW  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. z  e.  A  E. w  e.  A  z  ~<  w
) )
3534simp3bi 1005 . . . . . . . . . . . . . 14  |-  ( A  e.  InaccW  ->  A. z  e.  A  E. w  e.  A  z  ~<  w )
3635ad3antrrr 729 . . . . . . . . . . . . 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 4300 . . . . . . . . . . . . . . 15  |-  ( z  =  ( aleph `  y
)  ->  ( z  ~<  w  <->  ( aleph `  y
)  ~<  w ) )
3837rexbidv 2741 . . . . . . . . . . . . . 14  |-  ( z  =  ( aleph `  y
)  ->  ( E. w  e.  A  z  ~<  w  <->  E. w  e.  A  ( aleph `  y )  ~<  w ) )
3938rspcva 3076 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 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 615 . . . . . . . . . . 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 8150 . . . . . . . . . . . . . . . . . . 19  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. w  e.  A  w  ~<  A ) )
4342simprbi 464 . . . . . . . . . . . . . . . . . 18  |-  ( (
card `  A )  =  A  ->  A. w  e.  A  w  ~<  A )
44 rsp 2781 . . . . . . . . . . . . . . . . . 18  |-  ( A. w  e.  A  w  ~<  A  ->  ( w  e.  A  ->  w  ~<  A ) )
451, 43, 443syl 20 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  InaccW  ->  (
w  e.  A  ->  w  ~<  A ) )
4645ad3antrrr 729 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( w  e.  A  ->  w  ~<  A ) )
4732breq2d 4309 . . . . . . . . . . . . . . . 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 214 . . . . . . . . . . . . . . 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 8247 . . . . . . . . . . . . . . . 16  |-  -.  (
( aleph `  y )  ~<  w  /\  w  ~<  (
aleph `  suc  y ) )
50 pm3.21 448 . . . . . . . . . . . . . . . 16  |-  ( w 
~<  ( aleph `  suc  y )  ->  ( ( aleph `  y )  ~<  w  ->  ( ( aleph `  y
)  ~<  w  /\  w  ~<  ( aleph `  suc  y ) ) ) )
5149, 50mtoi 178 . . . . . . . . . . . . . . 15  |-  ( w 
~<  ( aleph `  suc  y )  ->  -.  ( aleph `  y )  ~<  w
)
5248, 51syl6 33 . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . 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 2824 . . . . . . . . . . . 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 615 . . . . . . . . . . 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 175 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  y  e.  On )  ->  -.  x  =  suc  y )
5756nrexdv 2824 . . . . . . . . 9  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  -.  E. y  e.  On  x  =  suc  y )
5857pm2.21d 106 . . . . . . . 8  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  ( E. y  e.  On  x  =  suc  y  ->  Lim  x ) )
59 simpr 461 . . . . . . . . 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 1282 . . . . . . 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 532 . . . . 5  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
( aleph `  x )  =  A  /\  Lim  x
) )
6463ex 434 . . . 4  |-  ( ( A  e.  InaccW  /\  A  =/=  om )  -> 
( ( x  e.  On  /\  A  =  ( aleph `  x )
)  ->  ( ( aleph `  x )  =  A  /\  Lim  x
) ) )
6564eximdv 1676 . . 3  |-  ( ( A  e.  InaccW  /\  A  =/=  om )  -> 
( E. x ( x  e.  On  /\  A  =  ( aleph `  x ) )  ->  E. x ( ( aleph `  x )  =  A  /\  Lim  x ) ) )
667, 65syl5bi 217 . 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 184    /\ wa 369    \/ w3o 964    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   _Vcvv 2977    C_ wss 3333   (/)c0 3642   class class class wbr 4297   Oncon0 4724   Lim wlim 4725   suc csuc 4726   ` cfv 5423   omcom 6481    ~< csdm 7314   cardccrd 8110   alephcale 8111   cfccf 8112   InaccWcwina 8854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-om 6482  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-oi 7729  df-har 7778  df-card 8114  df-aleph 8115  df-cf 8116  df-wina 8856
This theorem is referenced by:  winafp  8869
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