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Theorem r1tskina 8592
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 2554 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
2 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( R1 `  A )  e. 
Tarski )
3 simpll 731 . . . . . . . . . 10  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  A  e.  On )
4 onwf 7691 . . . . . . . . . . . . . . . 16  |-  On  C_  U. ( R1 " On )
54sseli 3289 . . . . . . . . . . . . . . 15  |-  ( A  e.  On  ->  A  e.  U. ( R1 " On ) )
6 eqid 2389 . . . . . . . . . . . . . . . 16  |-  ( rank `  A )  =  (
rank `  A )
7 rankr1c 7682 . . . . . . . . . . . . . . . 16  |-  ( A  e.  U. ( R1
" On )  -> 
( ( rank `  A
)  =  ( rank `  A )  <->  ( -.  A  e.  ( R1 `  ( rank `  A
) )  /\  A  e.  ( R1 `  suc  ( rank `  A )
) ) ) )
86, 7mpbii 203 . . . . . . . . . . . . . . 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 446 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
11 r1fnon 7628 . . . . . . . . . . . . . . . . 17  |-  R1  Fn  On
12 fndm 5486 . . . . . . . . . . . . . . . . 17  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1311, 12ax-mp 8 . . . . . . . . . . . . . . . 16  |-  dom  R1  =  On
1413eleq2i 2453 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom  R1  <->  A  e.  On )
15 rankonid 7690 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
1614, 15bitr3i 243 . . . . . . . . . . . . . 14  |-  ( A  e.  On  <->  ( rank `  A )  =  A )
17 fveq2 5670 . . . . . . . . . . . . . 14  |-  ( (
rank `  A )  =  A  ->  ( R1
`  ( rank `  A
) )  =  ( R1 `  A ) )
1816, 17sylbi 188 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( R1 `  ( rank `  A
) )  =  ( R1 `  A ) )
1910, 18neleqtrd 2484 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  -.  A  e.  ( R1 `  A ) )
2019adantl 453 . . . . . . . . . . 11  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  -.  A  e.  ( R1 `  A ) )
21 onssr1 7692 . . . . . . . . . . . . . 14  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )
2214, 21sylbir 205 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  A  C_  ( R1 `  A
) )
23 tsken 8564 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  C_  ( R1 `  A
) )  ->  ( A  ~~  ( R1 `  A )  \/  A  e.  ( R1 `  A
) ) )
2422, 23sylan2 461 . . . . . . . . . . . 12  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  ( A  ~~  ( R1 `  A )  \/  A  e.  ( R1 `  A
) ) )
2524ord 367 . . . . . . . . . . 11  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  ( -.  A  ~~  ( R1
`  A )  ->  A  e.  ( R1 `  A ) ) )
2620, 25mt3d 119 . . . . . . . . . 10  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  A  ~~  ( R1 `  A
) )
272, 3, 26syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  A  ~~  ( R1 `  A
) )
28 carden2b 7789 . . . . . . . . 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 444 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A  e.  On )
31 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  ( R1 `  A
)  e.  Tarski )
3222adantr 452 . . . . . . . . . . . . . 14  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A  C_  ( R1 `  A
) )
3332sselda 3293 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  e.  ( R1
`  A ) )
34 tsksdom 8566 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  x  e.  ( R1 `  A
) )  ->  x  ~<  ( R1 `  A
) )
3531, 33, 34syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  ~<  ( R1 `  A ) )
36 simpll 731 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  A  e.  On )
3726ensymd 7096 . . . . . . . . . . . . 13  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  ( R1 `  A )  ~~  A )
3831, 36, 37syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  ( R1 `  A
)  ~~  A )
39 sdomentr 7179 . . . . . . . . . . . 12  |-  ( ( x  ~<  ( R1 `  A )  /\  ( R1 `  A )  ~~  A )  ->  x  ~<  A )
4035, 38, 39syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  ~<  A )
4140ralrimiva 2734 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A. x  e.  A  x  ~<  A )
42 iscard 7797 . . . . . . . . . 10  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
4330, 41, 42sylanbrc 646 . . . . . . . . 9  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( card `  A )  =  A )
4443adantr 452 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  A )  =  A )
4529, 44eqtr3d 2423 . . . . . . 7  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  ( R1 `  A ) )  =  A )
46 r10 7629 . . . . . . . . . . 11  |-  ( R1
`  (/) )  =  (/)
47 on0eln0 4579 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
4847biimpar 472 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  (/)  e.  A
)
49 r1sdom 7635 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( R1 `  (/) )  ~< 
( R1 `  A
) )
5048, 49syldan 457 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  ( R1 `  (/) )  ~<  ( R1 `  A ) )
5146, 50syl5eqbrr 4189 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  (/)  ~<  ( R1 `  A ) )
52 fvex 5684 . . . . . . . . . . 11  |-  ( R1
`  A )  e. 
_V
53520sdom 7176 . . . . . . . . . 10  |-  ( (/)  ~< 
( R1 `  A
)  <->  ( R1 `  A )  =/=  (/) )
5451, 53sylib 189 . . . . . . . . 9  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  ( R1 `  A )  =/=  (/) )
5554adantlr 696 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( R1 `  A )  =/=  (/) )
56 tskcard 8591 . . . . . . . 8  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  ( R1 `  A )  =/=  (/) )  ->  ( card `  ( R1 `  A
) )  e.  Inacc )
572, 55, 56syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  ( R1 `  A ) )  e. 
Inacc )
5845, 57eqeltrrd 2464 . . . . . 6  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  A  e.  Inacc )
5958ex 424 . . . . 5  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( A  =/=  (/)  ->  A  e.  Inacc
) )
601, 59syl5bir 210 . . . 4  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( -.  A  =  (/)  ->  A  e.  Inacc ) )
6160orrd 368 . . 3  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  ( A  =  (/)  \/  A  e.  Inacc ) )
6261ex 424 . 2  |-  ( A  e.  On  ->  (
( R1 `  A
)  e.  Tarski  ->  ( A  =  (/)  \/  A  e.  Inacc ) ) )
63 fveq2 5670 . . . . 5  |-  ( A  =  (/)  ->  ( R1
`  A )  =  ( R1 `  (/) ) )
6463, 46syl6eq 2437 . . . 4  |-  ( A  =  (/)  ->  ( R1
`  A )  =  (/) )
65 0tsk 8565 . . . 4  |-  (/)  e.  Tarski
6664, 65syl6eqel 2477 . . 3  |-  ( A  =  (/)  ->  ( R1
`  A )  e. 
Tarski )
67 inatsk 8588 . . 3  |-  ( A  e.  Inacc  ->  ( R1 `  A )  e.  Tarski )
6866, 67jaoi 369 . 2  |-  ( ( A  =  (/)  \/  A  e.  Inacc )  ->  ( R1 `  A )  e. 
Tarski )
6962, 68impbid1 195 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 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651    C_ wss 3265   (/)c0 3573   U.cuni 3959   class class class wbr 4155   Oncon0 4524   suc csuc 4526   dom cdm 4820   "cima 4823    Fn wfn 5391   ` cfv 5396    ~~ cen 7044    ~< csdm 7046   R1cr1 7623   rankcrnk 7624   cardccrd 7757   Inacccina 8493   Tarskictsk 8558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-ac2 8278
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-smo 6546  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-oi 7414  df-har 7461  df-r1 7625  df-rank 7626  df-card 7761  df-aleph 7762  df-cf 7763  df-acn 7764  df-ac 7932  df-wina 8494  df-ina 8495  df-tsk 8559
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