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Theorem rankr1c 8300
Description: A relationship between the rank function and the cumulative hierarchy of sets function  R1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankr1c  |-  ( A  e.  U. ( R1
" On )  -> 
( B  =  (
rank `  A )  <->  ( -.  A  e.  ( R1 `  B )  /\  A  e.  ( R1 `  suc  B
) ) ) )

Proof of Theorem rankr1c
StepHypRef Expression
1 id 22 . . . 4  |-  ( B  =  ( rank `  A
)  ->  B  =  ( rank `  A )
)
2 rankdmr1 8280 . . . 4  |-  ( rank `  A )  e.  dom  R1
31, 2syl6eqel 2515 . . 3  |-  ( B  =  ( rank `  A
)  ->  B  e.  dom  R1 )
43a1i 11 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( B  =  (
rank `  A )  ->  B  e.  dom  R1 ) )
5 elfvdm 5907 . . . . 5  |-  ( A  e.  ( R1 `  suc  B )  ->  suc  B  e.  dom  R1 )
6 r1funlim 8245 . . . . . . 7  |-  ( Fun 
R1  /\  Lim  dom  R1 )
76simpri 463 . . . . . 6  |-  Lim  dom  R1
8 limsuc 6690 . . . . . 6  |-  ( Lim 
dom  R1  ->  ( B  e.  dom  R1  <->  suc  B  e. 
dom  R1 ) )
97, 8ax-mp 5 . . . . 5  |-  ( B  e.  dom  R1  <->  suc  B  e. 
dom  R1 )
105, 9sylibr 215 . . . 4  |-  ( A  e.  ( R1 `  suc  B )  ->  B  e.  dom  R1 )
1110adantl 467 . . 3  |-  ( ( -.  A  e.  ( R1 `  B )  /\  A  e.  ( R1 `  suc  B
) )  ->  B  e.  dom  R1 )
1211a1i 11 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( ( -.  A  e.  ( R1 `  B
)  /\  A  e.  ( R1 `  suc  B
) )  ->  B  e.  dom  R1 ) )
13 rankr1clem 8299 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( -.  A  e.  ( R1 `  B
)  <->  B  C_  ( rank `  A ) ) )
14 rankr1ag 8281 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  suc  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
159, 14sylan2b 477 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
16 rankon 8274 . . . . . . 7  |-  ( rank `  A )  e.  On
17 limord 5501 . . . . . . . . . 10  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
187, 17ax-mp 5 . . . . . . . . 9  |-  Ord  dom  R1
19 ordelon 5466 . . . . . . . . 9  |-  ( ( Ord  dom  R1  /\  B  e.  dom  R1 )  ->  B  e.  On )
2018, 19mpan 674 . . . . . . . 8  |-  ( B  e.  dom  R1  ->  B  e.  On )
2120adantl 467 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  B  e.  On )
22 onsssuc 5529 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  On  /\  B  e.  On )  ->  ( ( rank `  A
)  C_  B  <->  ( rank `  A )  e.  suc  B ) )
2316, 21, 22sylancr 667 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( rank `  A )  C_  B  <->  (
rank `  A )  e.  suc  B ) )
2415, 23bitr4d 259 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  C_  B )
)
2513, 24anbi12d 715 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( -.  A  e.  ( R1
`  B )  /\  A  e.  ( R1 ` 
suc  B ) )  <-> 
( B  C_  ( rank `  A )  /\  ( rank `  A )  C_  B ) ) )
26 eqss 3479 . . . 4  |-  ( B  =  ( rank `  A
)  <->  ( B  C_  ( rank `  A )  /\  ( rank `  A
)  C_  B )
)
2725, 26syl6rbbr 267 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( B  =  ( rank `  A
)  <->  ( -.  A  e.  ( R1 `  B
)  /\  A  e.  ( R1 `  suc  B
) ) ) )
2827ex 435 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( B  e.  dom  R1 
->  ( B  =  (
rank `  A )  <->  ( -.  A  e.  ( R1 `  B )  /\  A  e.  ( R1 `  suc  B
) ) ) ) )
294, 12, 28pm5.21ndd 355 1  |-  ( A  e.  U. ( R1
" On )  -> 
( B  =  (
rank `  A )  <->  ( -.  A  e.  ( R1 `  B )  /\  A  e.  ( R1 `  suc  B
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3436   U.cuni 4219   dom cdm 4853   "cima 4856   Ord word 5441   Oncon0 5442   Lim wlim 5443   suc csuc 5444   Fun wfun 5595   ` cfv 5601   R1cr1 8241   rankcrnk 8242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-r1 8243  df-rank 8244
This theorem is referenced by:  rankidn  8301  rankpwi  8302  rankr1g  8311  r1tskina  9214
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