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Theorem r1tskina 8945
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 2606 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 simplr 749 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( R1 `  A )  e. 
Tarski )
3 simpll 748 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  A  e.  On )
4 onwf 8033 . . . . . . . . . . . . . . . 16  |-  On  C_  U. ( R1 " On )
54sseli 3349 . . . . . . . . . . . . . . 15  |-  ( A  e.  On  ->  A  e.  U. ( R1 " On ) )
6 eqid 2441 . . . . . . . . . . . . . . . 16  |-  ( rank `  A )  =  (
rank `  A )
7 rankr1c 8024 . . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
11 r1fnon 7970 . . . . . . . . . . . . . . . . 17  |-  R1  Fn  On
12 fndm 5507 . . . . . . . . . . . . . . . . 17  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1311, 12ax-mp 5 . . . . . . . . . . . . . . . 16  |-  dom  R1  =  On
1413eleq2i 2505 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom  R1  <->  A  e.  On )
15 rankonid 8032 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
1614, 15bitr3i 251 . . . . . . . . . . . . . 14  |-  ( A  e.  On  <->  ( rank `  A )  =  A )
17 fveq2 5688 . . . . . . . . . . . . . 14  |-  ( (
rank `  A )  =  A  ->  ( R1
`  ( rank `  A
) )  =  ( R1 `  A ) )
1816, 17sylbi 195 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( R1 `  ( rank `  A
) )  =  ( R1 `  A ) )
1910, 18neleqtrd 2536 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  -.  A  e.  ( R1 `  A ) )
2019adantl 463 . . . . . . . . . . 11  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  -.  A  e.  ( R1 `  A ) )
21 onssr1 8034 . . . . . . . . . . . . . 14  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )
2214, 21sylbir 213 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  A  C_  ( R1 `  A
) )
23 tsken 8917 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  C_  ( R1 `  A
) )  ->  ( A  ~~  ( R1 `  A )  \/  A  e.  ( R1 `  A
) ) )
2422, 23sylan2 471 . . . . . . . . . . . 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 656 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  A  ~~  ( R1 `  A
) )
28 carden2b 8133 . . . . . . . . 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 454 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A  e.  On )
31 simplr 749 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  ( R1 `  A
)  e.  Tarski )
3222adantr 462 . . . . . . . . . . . . . 14  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A  C_  ( R1 `  A
) )
3332sselda 3353 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  e.  ( R1
`  A ) )
34 tsksdom 8919 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  x  e.  ( R1 `  A
) )  ->  x  ~<  ( R1 `  A
) )
3531, 33, 34syl2anc 656 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  ~<  ( R1 `  A ) )
36 simpll 748 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  A  e.  On )
3726ensymd 7356 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  ( R1 `  A )  ~~  A )
3831, 36, 37syl2anc 656 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  ( R1 `  A
)  ~~  A )
39 sdomentr 7441 . . . . . . . . . . . 12  |-  ( ( x  ~<  ( R1 `  A )  /\  ( R1 `  A )  ~~  A )  ->  x  ~<  A )
4035, 38, 39syl2anc 656 . . . . . . . . . . 11  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  ~<  A )
4140ralrimiva 2797 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A. x  e.  A  x  ~<  A )
42 iscard 8141 . . . . . . . . . 10  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
4330, 41, 42sylanbrc 659 . . . . . . . . 9  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( card `  A )  =  A )
4443adantr 462 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  A )  =  A )
4529, 44eqtr3d 2475 . . . . . . 7  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  ( R1 `  A ) )  =  A )
46 r10 7971 . . . . . . . . . . 11  |-  ( R1
`  (/) )  =  (/)
47 on0eln0 4770 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
4847biimpar 482 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  (/)  e.  A
)
49 r1sdom 7977 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( R1 `  (/) )  ~< 
( R1 `  A
) )
5048, 49syldan 467 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  ( R1 `  (/) )  ~<  ( R1 `  A ) )
5146, 50syl5eqbrr 4323 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  (/)  ~<  ( R1 `  A ) )
52 fvex 5698 . . . . . . . . . . 11  |-  ( R1
`  A )  e. 
_V
53520sdom 7438 . . . . . . . . . 10  |-  ( (/)  ~< 
( R1 `  A
)  <->  ( R1 `  A )  =/=  (/) )
5451, 53sylib 196 . . . . . . . . 9  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  ( R1 `  A )  =/=  (/) )
5554adantlr 709 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( R1 `  A )  =/=  (/) )
56 tskcard 8944 . . . . . . . 8  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  ( R1 `  A )  =/=  (/) )  ->  ( card `  ( R1 `  A
) )  e.  Inacc )
572, 55, 56syl2anc 656 . . . . . . 7  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  ( R1 `  A ) )  e. 
Inacc )
5845, 57eqeltrrd 2516 . . . . . 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 5688 . . . . 5  |-  ( A  =  (/)  ->  ( R1
`  A )  =  ( R1 `  (/) ) )
6463, 46syl6eq 2489 . . . 4  |-  ( A  =  (/)  ->  ( R1
`  A )  =  (/) )
65 0tsk 8918 . . . 4  |-  (/)  e.  Tarski
6664, 65syl6eqel 2529 . . 3  |-  ( A  =  (/)  ->  ( R1
`  A )  e. 
Tarski )
67 inatsk 8941 . . 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 1364    e. wcel 1761    =/= wne 2604   A.wral 2713    C_ wss 3325   (/)c0 3634   U.cuni 4088   class class class wbr 4289   Oncon0 4715   suc csuc 4717   dom cdm 4836   "cima 4839    Fn wfn 5410   ` cfv 5415    ~~ cen 7303    ~< csdm 7305   R1cr1 7965   rankcrnk 7966   cardccrd 8101   Inacccina 8846   Tarskictsk 8911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-ac2 8628
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-smo 6803  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-oi 7720  df-har 7769  df-r1 7967  df-rank 7968  df-card 8105  df-aleph 8106  df-cf 8107  df-acn 8108  df-ac 8282  df-wina 8847  df-ina 8848  df-tsk 8912
This theorem is referenced by: (None)
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