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Theorem r1tskina 8649
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 2600 . . . . 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 7748 . . . . . . . . . . . . . . . 16  |-  On  C_  U. ( R1 " On )
54sseli 3336 . . . . . . . . . . . . . . 15  |-  ( A  e.  On  ->  A  e.  U. ( R1 " On ) )
6 eqid 2435 . . . . . . . . . . . . . . . 16  |-  ( rank `  A )  =  (
rank `  A )
7 rankr1c 7739 . . . . . . . . . . . . . . . 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 7685 . . . . . . . . . . . . . . . . 17  |-  R1  Fn  On
12 fndm 5536 . . . . . . . . . . . . . . . . 17  |-  ( R1  Fn  On  ->  dom  R1  =  On )
1311, 12ax-mp 8 . . . . . . . . . . . . . . . 16  |-  dom  R1  =  On
1413eleq2i 2499 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom  R1  <->  A  e.  On )
15 rankonid 7747 . . . . . . . . . . . . . . 15  |-  ( A  e.  dom  R1  <->  ( rank `  A )  =  A )
1614, 15bitr3i 243 . . . . . . . . . . . . . 14  |-  ( A  e.  On  <->  ( rank `  A )  =  A )
17 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( (
rank `  A )  =  A  ->  ( R1
`  ( rank `  A
) )  =  ( R1 `  A ) )
1816, 17sylbi 188 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( R1 `  ( rank `  A
) )  =  ( R1 `  A ) )
1910, 18neleqtrd 2530 . . . . . . . . . . . 12  |-  ( A  e.  On  ->  -.  A  e.  ( R1 `  A ) )
2019adantl 453 . . . . . . . . . . 11  |-  ( ( ( R1 `  A
)  e.  Tarski  /\  A  e.  On )  ->  -.  A  e.  ( R1 `  A ) )
21 onssr1 7749 . . . . . . . . . . . . . 14  |-  ( A  e.  dom  R1  ->  A 
C_  ( R1 `  A ) )
2214, 21sylbir 205 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  A  C_  ( R1 `  A
) )
23 tsken 8621 . . . . . . . . . . . . 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 7846 . . . . . . . . 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 3340 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  x  e.  A )  ->  x  e.  ( R1
`  A ) )
34 tsksdom 8623 . . . . . . . . . . . . 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 7150 . . . . . . . . . . . . 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 7233 . . . . . . . . . . . 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 2781 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( R1 `  A )  e.  Tarski )  ->  A. x  e.  A  x  ~<  A )
42 iscard 7854 . . . . . . . . . 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 2469 . . . . . . 7  |-  ( ( ( A  e.  On  /\  ( R1 `  A
)  e.  Tarski )  /\  A  =/=  (/) )  ->  ( card `  ( R1 `  A ) )  =  A )
46 r10 7686 . . . . . . . . . . 11  |-  ( R1
`  (/) )  =  (/)
47 on0eln0 4628 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
4847biimpar 472 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  (/)  e.  A
)
49 r1sdom 7692 . . . . . . . . . . . 12  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( R1 `  (/) )  ~< 
( R1 `  A
) )
5048, 49syldan 457 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  ( R1 `  (/) )  ~<  ( R1 `  A ) )
5146, 50syl5eqbrr 4238 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  A  =/=  (/) )  ->  (/)  ~<  ( R1 `  A ) )
52 fvex 5734 . . . . . . . . . . 11  |-  ( R1
`  A )  e. 
_V
53520sdom 7230 . . . . . . . . . 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 8648 . . . . . . . 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 2510 . . . . . 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 5720 . . . . 5  |-  ( A  =  (/)  ->  ( R1
`  A )  =  ( R1 `  (/) ) )
6463, 46syl6eq 2483 . . . 4  |-  ( A  =  (/)  ->  ( R1
`  A )  =  (/) )
65 0tsk 8622 . . . 4  |-  (/)  e.  Tarski
6664, 65syl6eqel 2523 . . 3  |-  ( A  =  (/)  ->  ( R1
`  A )  e. 
Tarski )
67 inatsk 8645 . . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    C_ wss 3312   (/)c0 3620   U.cuni 4007   class class class wbr 4204   Oncon0 4573   suc csuc 4575   dom cdm 4870   "cima 4873    Fn wfn 5441   ` cfv 5446    ~~ cen 7098    ~< csdm 7100   R1cr1 7680   rankcrnk 7681   cardccrd 7814   Inacccina 8550   Tarskictsk 8615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-ac2 8335
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-smo 6600  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-har 7518  df-r1 7682  df-rank 7683  df-card 7818  df-aleph 7819  df-cf 7820  df-acn 7821  df-ac 7989  df-wina 8551  df-ina 8552  df-tsk 8616
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