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Theorem r1tskina 8961
Description: There is a direct relationship between transitive Tarski classes and inaccessible cardinals: the Tarski classes that occur in the cumulative hierarchy are exactly at the strongly inaccessible cardinals. (Contributed by Mario Carneiro, 8-Jun-2013.)
Assertion
Ref Expression
r1tskina  |-  ( A  e.  On  ->  (
( R1 `  A
)  e.  Tarski  <->  ( A  =  (/)  \/  A  e. 
Inacc ) ) )

Proof of Theorem r1tskina
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-ne 2620 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 simplr 754 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( R1 `  A )  e. 
Tarski )
3 simpll 753 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  A  e.  On )
4 onwf 8049 . . . . . . . . . . . . . . . 16  |-  On  C_  U. ( R1 " On )
54sseli 3364 . . . . . . . . . . . . . . 15  |-  ( A  e.  On  ->  A  e.  U. ( R1 " On ) )
6 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( rank `  A )  =  (
rank `  A )
7 rankr1c 8040 . . . . . . . . . . . . . . . 16  |-  ( A  e.  U. ( R1
" On )  -> 
( ( rank `  A
)  =  ( rank `  A )  <->  ( -.  A  e.  ( R1 `  ( rank `  A
) )  /\  A  e.  ( R1 `  suc  ( rank `  A )
) ) ) )
86, 7mpbii 211 . . . . . . . . . . . . . . 15  |-  ( A  e.  U. ( R1
" On )  -> 
( -.  A  e.  ( R1 `  ( rank `  A ) )  /\  A  e.  ( R1 `  suc  ( rank `  A ) ) ) )
95, 8syl 16 . . . . . . . . . . . . . 14  |-  ( A  e.  On  ->  ( -.  A  e.  ( R1 `  ( rank `  A
) )  /\  A  e.  ( R1 `  suc  ( rank `  A )
) ) )
109simpld 459 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
11 r1fnon 7986 . . . . . . . . . . . . . . . . 17  |-  R1  Fn  On
12 fndm 5522 . . . . . . . . . . . . . . . . 17  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1311, 12ax-mp 5 . . . . . . . . . . . . . . . 16  |-  dom  R1  =  On
1413eleq2i 2507 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom  R1  <->  A  e.  On )
15 rankonid 8048 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
1614, 15bitr3i 251 . . . . . . . . . . . . . 14  |-  ( A  e.  On  <->  ( rank `  A )  =  A )
17 fveq2 5703 . . . . . . . . . . . . . 14  |-  ( (
rank `  A )  =  A  ->  ( R1
`  ( rank `  A
) )  =  ( R1 `  A ) )
1816, 17sylbi 195 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( R1 `  ( rank `  A
) )  =  ( R1 `  A ) )
1910, 18neleqtrd 2539 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  -.  A  e.  ( R1 `  A ) )
2019adantl 466 . . . . . . . . . . 11  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  -.  A  e.  ( R1 `  A ) )
21 onssr1 8050 . . . . . . . . . . . . . 14  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )
2214, 21sylbir 213 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  A  C_  ( R1 `  A
) )
23 tsken 8933 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  C_  ( R1 `  A
) )  ->  ( A  ~~  ( R1 `  A )  \/  A  e.  ( R1 `  A
) ) )
2422, 23sylan2 474 . . . . . . . . . . . 12  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  ( A  ~~  ( R1 `  A )  \/  A  e.  ( R1 `  A
) ) )
2524ord 377 . . . . . . . . . . 11  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  ( -.  A  ~~  ( R1
`  A )  ->  A  e.  ( R1 `  A ) ) )
2620, 25mt3d 125 . . . . . . . . . 10  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  A  ~~  ( R1 `  A
) )
272, 3, 26syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  A  ~~  ( R1 `  A
) )
28 carden2b 8149 . . . . . . . . 9  |-  ( A 
~~  ( R1 `  A )  ->  ( card `  A )  =  ( card `  ( R1 `  A ) ) )
2927, 28syl 16 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  A )  =  ( card `  ( R1 `  A ) ) )
30 simpl 457 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A  e.  On )
31 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  ( R1 `  A
)  e.  Tarski )
3222adantr 465 . . . . . . . . . . . . . 14  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A  C_  ( R1 `  A
) )
3332sselda 3368 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  e.  ( R1
`  A ) )
34 tsksdom 8935 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  x  e.  ( R1 `  A
) )  ->  x  ~<  ( R1 `  A
) )
3531, 33, 34syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  ~<  ( R1 `  A ) )
36 simpll 753 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  A  e.  On )
3726ensymd 7372 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  ( R1 `  A )  ~~  A )
3831, 36, 37syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  ( R1 `  A
)  ~~  A )
39 sdomentr 7457 . . . . . . . . . . . 12  |-  ( ( x  ~<  ( R1 `  A )  /\  ( R1 `  A )  ~~  A )  ->  x  ~<  A )
4035, 38, 39syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  ~<  A )
4140ralrimiva 2811 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A. x  e.  A  x  ~<  A )
42 iscard 8157 . . . . . . . . . 10  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
4330, 41, 42sylanbrc 664 . . . . . . . . 9  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( card `  A )  =  A )
4443adantr 465 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  A )  =  A )
4529, 44eqtr3d 2477 . . . . . . 7  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  ( R1 `  A ) )  =  A )
46 r10 7987 . . . . . . . . . . 11  |-  ( R1
`  (/) )  =  (/)
47 on0eln0 4786 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
4847biimpar 485 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  (/)  e.  A
)
49 r1sdom 7993 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( R1 `  (/) )  ~< 
( R1 `  A
) )
5048, 49syldan 470 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  ( R1 `  (/) )  ~<  ( R1 `  A ) )
5146, 50syl5eqbrr 4338 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  (/)  ~<  ( R1 `  A ) )
52 fvex 5713 . . . . . . . . . . 11  |-  ( R1
`  A )  e. 
_V
53520sdom 7454 . . . . . . . . . 10  |-  ( (/)  ~< 
( R1 `  A
)  <->  ( R1 `  A )  =/=  (/) )
5451, 53sylib 196 . . . . . . . . 9  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  ( R1 `  A )  =/=  (/) )
5554adantlr 714 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( R1 `  A )  =/=  (/) )
56 tskcard 8960 . . . . . . . 8  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  ( R1 `  A )  =/=  (/) )  ->  ( card `  ( R1 `  A
) )  e.  Inacc )
572, 55, 56syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  ( R1 `  A ) )  e. 
Inacc )
5845, 57eqeltrrd 2518 . . . . . 6  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  A  e.  Inacc )
5958ex 434 . . . . 5  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( A  =/=  (/)  ->  A  e.  Inacc
) )
601, 59syl5bir 218 . . . 4  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( -.  A  =  (/)  ->  A  e.  Inacc ) )
6160orrd 378 . . 3  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( A  =  (/)  \/  A  e.  Inacc ) )
6261ex 434 . 2  |-  ( A  e.  On  ->  (
( R1 `  A
)  e.  Tarski  ->  ( A  =  (/)  \/  A  e.  Inacc ) ) )
63 fveq2 5703 . . . . 5  |-  ( A  =  (/)  ->  ( R1
`  A )  =  ( R1 `  (/) ) )
6463, 46syl6eq 2491 . . . 4  |-  ( A  =  (/)  ->  ( R1
`  A )  =  (/) )
65 0tsk 8934 . . . 4  |-  (/)  e.  Tarski
6664, 65syl6eqel 2531 . . 3  |-  ( A  =  (/)  ->  ( R1
`  A )  e. 
Tarski )
67 inatsk 8957 . . 3  |-  ( A  e.  Inacc  ->  ( R1 `  A )  e.  Tarski )
6866, 67jaoi 379 . 2  |-  ( ( A  =  (/)  \/  A  e.  Inacc )  ->  ( R1 `  A )  e. 
Tarski )
6962, 68impbid1 203 1  |-  ( A  e.  On  ->  (
( R1 `  A
)  e.  Tarski  <->  ( A  =  (/)  \/  A  e. 
Inacc ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727    C_ wss 3340   (/)c0 3649   U.cuni 4103   class class class wbr 4304   Oncon0 4731   suc csuc 4733   dom cdm 4852   "cima 4855    Fn wfn 5425   ` cfv 5430    ~~ cen 7319    ~< csdm 7321   R1cr1 7981   rankcrnk 7982   cardccrd 8117   Inacccina 8862   Tarskictsk 8927
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-ac2 8644
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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-smo 6819  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-oi 7736  df-har 7785  df-r1 7983  df-rank 7984  df-card 8121  df-aleph 8122  df-cf 8123  df-acn 8124  df-ac 8298  df-wina 8863  df-ina 8864  df-tsk 8928
This theorem is referenced by: (None)
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