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Theorem winalim2 9063
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 9059 . . . 4  |-  ( A  e.  InaccW  ->  ( card `  A )  =  A )
2 winainf 9061 . . . . 5  |-  ( A  e.  InaccW  ->  om  C_  A
)
3 cardalephex 8462 . . . . 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 463 . 2  |-  ( ( A  e.  InaccW  /\  A  =/=  om )  ->  E. x  e.  On  A  =  ( aleph `  x ) )
7 df-rex 2810 . . 3  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  <->  E. x ( x  e.  On  /\  A  =  ( aleph `  x
) ) )
8 simprr 755 . . . . . . 7  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  A  =  ( aleph `  x
) )
98eqcomd 2462 . . . . . 6  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  ( aleph `  x )  =  A )
10 simprl 754 . . . . . . . 8  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  x  e.  On )
11 onzsl 6654 . . . . . . . 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 753 . . . . . . . . . 10  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  A  =/=  om )
14 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
15 aleph0 8438 . . . . . . . . . . . . . 14  |-  ( aleph `  (/) )  =  om
1614, 15syl6eq 2511 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  ( aleph `  x )  =  om )
17 eqtr 2480 . . . . . . . . . . . . 13  |-  ( ( A  =  ( aleph `  x )  /\  ( aleph `  x )  =  om )  ->  A  =  om )
1816, 17sylan2 472 . . . . . . . . . . . 12  |-  ( ( A  =  ( aleph `  x )  /\  x  =  (/) )  ->  A  =  om )
1918ex 432 . . . . . . . . . . 11  |-  ( A  =  ( aleph `  x
)  ->  ( x  =  (/)  ->  A  =  om ) )
2019necon3ad 2664 . . . . . . . . . 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 6621 . . . . . . . . . . . . . . . 16  |-  ( y  e.  On  ->  suc  y  e.  On )
24 vex 3109 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
2524sucid 4946 . . . . . . . . . . . . . . . 16  |-  y  e. 
suc  y
26 alephord2i 8449 . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 5848 . . . . . . . . . . . . . . . 16  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
3130ad2antll 726 . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  A  =  (
aleph `  suc  y ) )
3328, 32eleqtrrd 2545 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( aleph `  y
)  e.  A )
34 elwina 9053 . . . . . . . . . . . . . . 15  |-  ( A  e.  InaccW  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. z  e.  A  E. w  e.  A  z  ~<  w
) )
3534simp3bi 1011 . . . . . . . . . . . . . 14  |-  ( A  e.  InaccW  ->  A. z  e.  A  E. w  e.  A  z  ~<  w )
3635ad3antrrr 727 . . . . . . . . . . . . 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 4442 . . . . . . . . . . . . . . 15  |-  ( z  =  ( aleph `  y
)  ->  ( z  ~<  w  <->  ( aleph `  y
)  ~<  w ) )
3837rexbidv 2965 . . . . . . . . . . . . . 14  |-  ( z  =  ( aleph `  y
)  ->  ( E. w  e.  A  z  ~<  w  <->  E. w  e.  A  ( aleph `  y )  ~<  w ) )
3938rspcva 3205 . . . . . . . . . . . . 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 659 . . . . . . . . . . . 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 613 . . . . . . . . . . 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 8347 . . . . . . . . . . . . . . . . . . 19  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. w  e.  A  w  ~<  A ) )
4342simprbi 462 . . . . . . . . . . . . . . . . . 18  |-  ( (
card `  A )  =  A  ->  A. w  e.  A  w  ~<  A )
44 rsp 2820 . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e. 
InaccW  /\  A  =/= 
om )  /\  (
x  e.  On  /\  A  =  ( aleph `  x ) ) )  /\  ( y  e.  On  /\  x  =  suc  y ) )  ->  ( w  e.  A  ->  w  ~<  A ) )
4732breq2d 4451 . . . . . . . . . . . . . . . 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 8444 . . . . . . . . . . . . . . . 16  |-  -.  (
( aleph `  y )  ~<  w  /\  w  ~<  (
aleph `  suc  y ) )
50 pm3.21 446 . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . 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 2910 . . . . . . . . . . . 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 613 . . . . . . . . . . 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 2910 . . . . . . . . 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 459 . . . . . . . . 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 1290 . . . . . . 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 530 . . . . 5  |-  ( ( ( A  e.  InaccW  /\  A  =/=  om )  /\  ( x  e.  On  /\  A  =  ( aleph `  x )
) )  ->  (
( aleph `  x )  =  A  /\  Lim  x
) )
6463ex 432 . . . 4  |-  ( ( A  e.  InaccW  /\  A  =/=  om )  -> 
( ( x  e.  On  /\  A  =  ( aleph `  x )
)  ->  ( ( aleph `  x )  =  A  /\  Lim  x
) ) )
6564eximdv 1715 . . 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 367    \/ w3o 970    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106    C_ wss 3461   (/)c0 3783   class class class wbr 4439   Oncon0 4867   Lim wlim 4868   suc csuc 4869   ` cfv 5570   omcom 6673    ~< csdm 7508   cardccrd 8307   alephcale 8308   cfccf 8309   InaccWcwina 9049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-har 7976  df-card 8311  df-aleph 8312  df-cf 8313  df-wina 9051
This theorem is referenced by:  winafp  9064
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