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Theorem r1tskina 8420
Description: There is a direct relationship between transitive Tarski's classes and inacessible cardinals: the Tarski's 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 2461 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( R1 `  A )  e. 
Tarski )
3 simpll 730 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  A  e.  On )
4 onwf 7518 . . . . . . . . . . . . . . . 16  |-  On  C_  U. ( R1 " On )
54sseli 3189 . . . . . . . . . . . . . . 15  |-  ( A  e.  On  ->  A  e.  U. ( R1 " On ) )
6 eqid 2296 . . . . . . . . . . . . . . . 16  |-  ( rank `  A )  =  (
rank `  A )
7 rankr1c 7509 . . . . . . . . . . . . . . . 16  |-  ( A  e.  U. ( R1
" On )  -> 
( ( rank `  A
)  =  ( rank `  A )  <->  ( -.  A  e.  ( R1 `  ( rank `  A
) )  /\  A  e.  ( R1 `  suc  ( rank `  A )
) ) ) )
86, 7mpbii 202 . . . . . . . . . . . . . . 15  |-  ( A  e.  U. ( R1
" On )  -> 
( -.  A  e.  ( R1 `  ( rank `  A ) )  /\  A  e.  ( R1 `  suc  ( rank `  A ) ) ) )
95, 8syl 15 . . . . . . . . . . . . . 14  |-  ( A  e.  On  ->  ( -.  A  e.  ( R1 `  ( rank `  A
) )  /\  A  e.  ( R1 `  suc  ( rank `  A )
) ) )
109simpld 445 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
11 r1fnon 7455 . . . . . . . . . . . . . . . . . 18  |-  R1  Fn  On
12 fndm 5359 . . . . . . . . . . . . . . . . . 18  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1311, 12ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  dom  R1  =  On
1413eleq2i 2360 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom  R1  <->  A  e.  On )
15 rankonid 7517 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
1614, 15bitr3i 242 . . . . . . . . . . . . . . 15  |-  ( A  e.  On  <->  ( rank `  A )  =  A )
17 fveq2 5541 . . . . . . . . . . . . . . 15  |-  ( (
rank `  A )  =  A  ->  ( R1
`  ( rank `  A
) )  =  ( R1 `  A ) )
1816, 17sylbi 187 . . . . . . . . . . . . . 14  |-  ( A  e.  On  ->  ( R1 `  ( rank `  A
) )  =  ( R1 `  A ) )
1918eleq2d 2363 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( A  e.  ( R1 `  ( rank `  A
) )  <->  A  e.  ( R1 `  A ) ) )
2010, 19mtbid 291 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  -.  A  e.  ( R1 `  A ) )
2120adantl 452 . . . . . . . . . . 11  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  -.  A  e.  ( R1 `  A ) )
22 onssr1 7519 . . . . . . . . . . . . . 14  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )
2314, 22sylbir 204 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  A  C_  ( R1 `  A
) )
24 tsken 8392 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  C_  ( R1 `  A
) )  ->  ( A  ~~  ( R1 `  A )  \/  A  e.  ( R1 `  A
) ) )
2523, 24sylan2 460 . . . . . . . . . . . 12  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  ( A  ~~  ( R1 `  A )  \/  A  e.  ( R1 `  A
) ) )
2625ord 366 . . . . . . . . . . 11  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  ( -.  A  ~~  ( R1
`  A )  ->  A  e.  ( R1 `  A ) ) )
2721, 26mt3d 117 . . . . . . . . . 10  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  A  ~~  ( R1 `  A
) )
282, 3, 27syl2anc 642 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  A  ~~  ( R1 `  A
) )
29 carden2b 7616 . . . . . . . . 9  |-  ( A 
~~  ( R1 `  A )  ->  ( card `  A )  =  ( card `  ( R1 `  A ) ) )
3028, 29syl 15 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  A )  =  ( card `  ( R1 `  A ) ) )
31 simpl 443 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A  e.  On )
32 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  ( R1 `  A
)  e.  Tarski )
3323adantr 451 . . . . . . . . . . . . . 14  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A  C_  ( R1 `  A
) )
3433sselda 3193 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  e.  ( R1
`  A ) )
35 tsksdom 8394 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  x  e.  ( R1 `  A
) )  ->  x  ~<  ( R1 `  A
) )
3632, 34, 35syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  ~<  ( R1 `  A ) )
37 simpll 730 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  A  e.  On )
38 ensym 6926 . . . . . . . . . . . . . 14  |-  ( A 
~~  ( R1 `  A )  ->  ( R1 `  A )  ~~  A )
3927, 38syl 15 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  ( R1 `  A )  ~~  A )
4032, 37, 39syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  ( R1 `  A
)  ~~  A )
41 sdomentr 7011 . . . . . . . . . . . 12  |-  ( ( x  ~<  ( R1 `  A )  /\  ( R1 `  A )  ~~  A )  ->  x  ~<  A )
4236, 40, 41syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  ~<  A )
4342ralrimiva 2639 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A. x  e.  A  x  ~<  A )
44 iscard 7624 . . . . . . . . . 10  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
4531, 43, 44sylanbrc 645 . . . . . . . . 9  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( card `  A )  =  A )
4645adantr 451 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  A )  =  A )
4730, 46eqtr3d 2330 . . . . . . 7  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  ( R1 `  A ) )  =  A )
48 r10 7456 . . . . . . . . . . 11  |-  ( R1
`  (/) )  =  (/)
49 on0eln0 4463 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
5049biimpar 471 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  (/)  e.  A
)
51 r1sdom 7462 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( R1 `  (/) )  ~< 
( R1 `  A
) )
5250, 51syldan 456 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  ( R1 `  (/) )  ~<  ( R1 `  A ) )
5348, 52syl5eqbrr 4073 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  (/)  ~<  ( R1 `  A ) )
54 fvex 5555 . . . . . . . . . . 11  |-  ( R1
`  A )  e. 
_V
55540sdom 7008 . . . . . . . . . 10  |-  ( (/)  ~< 
( R1 `  A
)  <->  ( R1 `  A )  =/=  (/) )
5653, 55sylib 188 . . . . . . . . 9  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  ( R1 `  A )  =/=  (/) )
5756adantlr 695 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( R1 `  A )  =/=  (/) )
58 tskcard 8419 . . . . . . . 8  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  ( R1 `  A )  =/=  (/) )  ->  ( card `  ( R1 `  A
) )  e.  Inacc )
592, 57, 58syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  ( R1 `  A ) )  e. 
Inacc )
6047, 59eqeltrrd 2371 . . . . . 6  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  A  e.  Inacc )
6160ex 423 . . . . 5  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( A  =/=  (/)  ->  A  e.  Inacc
) )
621, 61syl5bir 209 . . . 4  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( -.  A  =  (/)  ->  A  e.  Inacc ) )
6362orrd 367 . . 3  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( A  =  (/)  \/  A  e.  Inacc ) )
6463ex 423 . 2  |-  ( A  e.  On  ->  (
( R1 `  A
)  e.  Tarski  ->  ( A  =  (/)  \/  A  e.  Inacc ) ) )
65 fveq2 5541 . . . . 5  |-  ( A  =  (/)  ->  ( R1
`  A )  =  ( R1 `  (/) ) )
6665, 48syl6eq 2344 . . . 4  |-  ( A  =  (/)  ->  ( R1
`  A )  =  (/) )
67 0tsk 8393 . . . 4  |-  (/)  e.  Tarski
6866, 67syl6eqel 2384 . . 3  |-  ( A  =  (/)  ->  ( R1
`  A )  e. 
Tarski )
69 inatsk 8416 . . 3  |-  ( A  e.  Inacc  ->  ( R1 `  A )  e.  Tarski )
7068, 69jaoi 368 . 2  |-  ( ( A  =  (/)  \/  A  e.  Inacc )  ->  ( R1 `  A )  e. 
Tarski )
7164, 70impbid1 194 1  |-  ( A  e.  On  ->  (
( R1 `  A
)  e.  Tarski  <->  ( A  =  (/)  \/  A  e. 
Inacc ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   U.cuni 3843   class class class wbr 4039   Oncon0 4408   suc csuc 4410   dom cdm 4705   "cima 4708    Fn wfn 5266   ` cfv 5271    ~~ cen 6876    ~< csdm 6878   R1cr1 7450   rankcrnk 7451   cardccrd 7584   Inacccina 8321   Tarskictsk 8386
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  ax-ac2 8105
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-iin 3924  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-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-smo 6379  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-r1 7452  df-rank 7453  df-card 7588  df-aleph 7589  df-cf 7590  df-acn 7591  df-ac 7759  df-wina 8322  df-ina 8323  df-tsk 8387
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