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Theorem winalim2 8334
Description: A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)
Assertion
Ref Expression
winalim2  |-  ( ( A  e.  Inacc W  /\  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 8330 . . . 4  |-  ( A  e.  Inacc W  ->  ( card `  A )  =  A )
2 winainf 8332 . . . . 5  |-  ( A  e.  Inacc W  ->  om  C_  A
)
3 cardalephex 7733 . . . . 5  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
42, 3syl 15 . . . 4  |-  ( A  e.  Inacc W  ->  (
( card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
51, 4mpbid 201 . . 3  |-  ( A  e.  Inacc W  ->  E. x  e.  On  A  =  (
aleph `  x ) )
65adantr 451 . 2  |-  ( ( A  e.  Inacc W  /\  A  =/=  om )  ->  E. x  e.  On  A  =  ( aleph `  x ) )
7 df-rex 2562 . . 3  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  <->  E. x ( x  e.  On  /\  A  =  ( aleph `  x
) ) )
8 simprr 733 . . . . . . 7  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  A  =  ( aleph `  x
) )
98eqcomd 2301 . . . . . 6  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  ( aleph `  x )  =  A )
10 simprl 732 . . . . . . . 8  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  x  e.  On )
11 onzsl 4653 . . . . . . . 8  |-  ( x  e.  On  <->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y  \/  ( x  e.  _V  /\  Lim  x
) ) )
1210, 11sylib 188 . . . . . . 7  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
x  =  (/)  \/  E. y  e.  On  x  =  suc  y  \/  (
x  e.  _V  /\  Lim  x ) ) )
13 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  A  =/=  om )
14 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
15 aleph0 7709 . . . . . . . . . . . . . 14  |-  ( aleph `  (/) )  =  om
1614, 15syl6eq 2344 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( aleph `  x )  =  om )
17 eqtr 2313 . . . . . . . . . . . . 13  |-  ( ( A  =  ( aleph `  x )  /\  ( aleph `  x )  =  om )  ->  A  =  om )
1816, 17sylan2 460 . . . . . . . . . . . 12  |-  ( ( A  =  ( aleph `  x )  /\  x  =  (/) )  ->  A  =  om )
1918ex 423 . . . . . . . . . . 11  |-  ( A  =  ( aleph `  x
)  ->  ( x  =  (/)  ->  A  =  om ) )
2019necon3ad 2495 . . . . . . . . . 10  |-  ( A  =  ( aleph `  x
)  ->  ( A  =/=  om  ->  -.  x  =  (/) ) )
218, 13, 20sylc 56 . . . . . . . . 9  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  -.  x  =  (/) )
2221pm2.21d 98 . . . . . . . 8  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
x  =  (/)  ->  Lim  x ) )
23 suceloni 4620 . . . . . . . . . . . . . . . 16  |-  ( y  e.  On  ->  suc  y  e.  On )
24 vex 2804 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
2524sucid 4487 . . . . . . . . . . . . . . . 16  |-  y  e. 
suc  y
26 alephord2i 7720 . . . . . . . . . . . . . . . 16  |-  ( suc  y  e.  On  ->  ( y  e.  suc  y  ->  ( aleph `  y )  e.  ( aleph `  suc  y ) ) )
2723, 25, 26ee10 1366 . . . . . . . . . . . . . . 15  |-  ( y  e.  On  ->  ( aleph `  y )  e.  ( aleph `  suc  y ) )
2827ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( aleph `  y
)  e.  ( aleph ` 
suc  y ) )
29 simplrr 737 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  A  =  (
aleph `  x ) )
30 fveq2 5541 . . . . . . . . . . . . . . . 16  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
3130ad2antll 709 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( aleph `  x
)  =  ( aleph ` 
suc  y ) )
3229, 31eqtrd 2328 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  A  =  (
aleph `  suc  y ) )
3328, 32eleqtrrd 2373 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( aleph `  y
)  e.  A )
34 elwina 8324 . . . . . . . . . . . . . . 15  |-  ( A  e.  Inacc W  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. z  e.  A  E. w  e.  A  z  ~<  w
) )
3534simp3bi 972 . . . . . . . . . . . . . 14  |-  ( A  e.  Inacc W  ->  A. z  e.  A  E. w  e.  A  z  ~<  w )
3635ad3antrrr 710 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
Inacc W  /\  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 4042 . . . . . . . . . . . . . . 15  |-  ( z  =  ( aleph `  y
)  ->  ( z  ~<  w  <->  ( aleph `  y
)  ~<  w ) )
3837rexbidv 2577 . . . . . . . . . . . . . 14  |-  ( z  =  ( aleph `  y
)  ->  ( E. w  e.  A  z  ~<  w  <->  E. w  e.  A  ( aleph `  y )  ~<  w ) )
3938rspcva 2895 . . . . . . . . . . . . 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 642 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  E. w  e.  A  ( aleph `  y )  ~<  w )
4140expr 598 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  y  e.  On )  ->  ( x  =  suc  y  ->  E. w  e.  A  ( aleph `  y )  ~<  w
) )
42 iscard 7624 . . . . . . . . . . . . . . . . . . 19  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. w  e.  A  w  ~<  A ) )
4342simprbi 450 . . . . . . . . . . . . . . . . . 18  |-  ( (
card `  A )  =  A  ->  A. w  e.  A  w  ~<  A )
44 rsp 2616 . . . . . . . . . . . . . . . . . 18  |-  ( A. w  e.  A  w  ~<  A  ->  ( w  e.  A  ->  w  ~<  A ) )
451, 43, 443syl 18 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  Inacc W  ->  (
w  e.  A  ->  w  ~<  A ) )
4645ad3antrrr 710 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( w  e.  A  ->  w  ~<  A ) )
4732breq2d 4051 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( w  ~<  A  <-> 
w  ~<  ( aleph `  suc  y ) ) )
4846, 47sylibd 205 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( w  e.  A  ->  w  ~<  (
aleph `  suc  y ) ) )
49 alephnbtwn2 7715 . . . . . . . . . . . . . . . 16  |-  -.  (
( aleph `  y )  ~<  w  /\  w  ~<  (
aleph `  suc  y ) )
50 pm3.21 435 . . . . . . . . . . . . . . . 16  |-  ( w 
~<  ( aleph `  suc  y )  ->  ( ( aleph `  y )  ~<  w  ->  ( ( aleph `  y
)  ~<  w  /\  w  ~<  ( aleph `  suc  y ) ) ) )
5149, 50mtoi 169 . . . . . . . . . . . . . . 15  |-  ( w 
~<  ( aleph `  suc  y )  ->  -.  ( aleph `  y )  ~<  w
)
5248, 51syl6 29 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( w  e.  A  ->  -.  ( aleph `  y )  ~<  w ) )
5352imp 418 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  /\  w  e.  A
)  ->  -.  ( aleph `  y )  ~<  w )
5453nrexdv 2659 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  -.  E. w  e.  A  ( aleph `  y )  ~<  w
)
5554expr 598 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  y  e.  On )  ->  ( x  =  suc  y  ->  -.  E. w  e.  A  (
aleph `  y )  ~<  w ) )
5641, 55pm2.65d 166 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
Inacc W  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  y  e.  On )  ->  -.  x  =  suc  y )
5756nrexdv 2659 . . . . . . . . 9  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  -.  E. y  e.  On  x  =  suc  y )
5857pm2.21d 98 . . . . . . . 8  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  ( E. y  e.  On  x  =  suc  y  ->  Lim  x ) )
59 simpr 447 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  Lim  x )  ->  Lim  x )
6059a1i 10 . . . . . . . 8  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
( x  e.  _V  /\ 
Lim  x )  ->  Lim  x ) )
6122, 58, 603jaod 1246 . . . . . . 7  |-  ( ( ( A  e.  Inacc W  /\  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 14 . . . . . 6  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  Lim  x )
639, 62jca 518 . . . . 5  |-  ( ( ( A  e.  Inacc W  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
( aleph `  x )  =  A  /\  Lim  x
) )
6463ex 423 . . . 4  |-  ( ( A  e.  Inacc W  /\  A  =/=  om )  -> 
( ( x  e.  On  /\  A  =  ( aleph `  x )
)  ->  ( ( aleph `  x )  =  A  /\  Lim  x
) ) )
6564eximdv 1612 . . 3  |-  ( ( A  e.  Inacc W  /\  A  =/=  om )  -> 
( E. x ( x  e.  On  /\  A  =  ( aleph `  x ) )  ->  E. x ( ( aleph `  x )  =  A  /\  Lim  x ) ) )
667, 65syl5bi 208 . 2  |-  ( ( A  e.  Inacc W  /\  A  =/=  om )  -> 
( E. x  e.  On  A  =  (
aleph `  x )  ->  E. x ( ( aleph `  x )  =  A  /\  Lim  x ) ) )
676, 66mpd 14 1  |-  ( ( A  e.  Inacc W  /\  A  =/=  om )  ->  E. x ( ( aleph `  x )  =  A  /\  Lim  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   (/)c0 3468   class class class wbr 4039   Oncon0 4408   Lim wlim 4409   suc csuc 4410   omcom 4672   ` cfv 5271    ~< csdm 6878   cardccrd 7584   alephcale 7585   cfccf 7586   Inacc Wcwina 8320
This theorem is referenced by:  winafp  8335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589  df-cf 7590  df-wina 8322
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