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Theorem rankr1c 7489
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 19 . . . 4  |-  ( B  =  ( rank `  A
)  ->  B  =  ( rank `  A )
)
2 rankdmr1 7469 . . . 4  |-  ( rank `  A )  e.  dom  R1
31, 2syl6eqel 2372 . . 3  |-  ( B  =  ( rank `  A
)  ->  B  e.  dom  R1 )
43a1i 10 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( B  =  (
rank `  A )  ->  B  e.  dom  R1 ) )
5 elfvdm 5516 . . . . 5  |-  ( A  e.  ( R1 `  suc  B )  ->  suc  B  e.  dom  R1 )
6 r1funlim 7434 . . . . . . 7  |-  ( Fun 
R1  /\  Lim  dom  R1 )
76simpri 448 . . . . . 6  |-  Lim  dom  R1
8 limsuc 4639 . . . . . 6  |-  ( Lim 
dom  R1  ->  ( B  e.  dom  R1  <->  suc  B  e. 
dom  R1 ) )
97, 8ax-mp 8 . . . . 5  |-  ( B  e.  dom  R1  <->  suc  B  e. 
dom  R1 )
105, 9sylibr 203 . . . 4  |-  ( A  e.  ( R1 `  suc  B )  ->  B  e.  dom  R1 )
1110adantl 452 . . 3  |-  ( ( -.  A  e.  ( R1 `  B )  /\  A  e.  ( R1 `  suc  B
) )  ->  B  e.  dom  R1 )
1211a1i 10 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( ( -.  A  e.  ( R1 `  B
)  /\  A  e.  ( R1 `  suc  B
) )  ->  B  e.  dom  R1 ) )
13 rankr1clem 7488 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( -.  A  e.  ( R1 `  B
)  <->  B  C_  ( rank `  A ) ) )
14 rankr1ag 7470 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  suc  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
159, 14sylan2b 461 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
16 rankon 7463 . . . . . . 7  |-  ( rank `  A )  e.  On
17 limord 4450 . . . . . . . . . 10  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
187, 17ax-mp 8 . . . . . . . . 9  |-  Ord  dom  R1
19 ordelon 4415 . . . . . . . . 9  |-  ( ( Ord  dom  R1  /\  B  e.  dom  R1 )  ->  B  e.  On )
2018, 19mpan 651 . . . . . . . 8  |-  ( B  e.  dom  R1  ->  B  e.  On )
2120adantl 452 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  B  e.  On )
22 onsssuc 4479 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  On  /\  B  e.  On )  ->  ( ( rank `  A
)  C_  B  <->  ( rank `  A )  e.  suc  B ) )
2316, 21, 22sylancr 644 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( rank `  A )  C_  B  <->  (
rank `  A )  e.  suc  B ) )
2415, 23bitr4d 247 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  C_  B )
)
2513, 24anbi12d 691 . . . 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 3195 . . . 4  |-  ( B  =  ( rank `  A
)  <->  ( B  C_  ( rank `  A )  /\  ( rank `  A
)  C_  B )
)
2725, 26syl6rbbr 255 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( B  =  ( rank `  A
)  <->  ( -.  A  e.  ( R1 `  B
)  /\  A  e.  ( R1 `  suc  B
) ) ) )
2827ex 423 . 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 343 1  |-  ( A  e.  U. ( R1
" On )  -> 
( B  =  (
rank `  A )  <->  ( -.  A  e.  ( R1 `  B )  /\  A  e.  ( R1 `  suc  B
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685    C_ wss 3153   U.cuni 3828   Ord word 4390   Oncon0 4391   Lim wlim 4392   suc csuc 4393    dom cdm 4688   "cima 4691   Fun wfun 5215   ` cfv 5221   R1cr1 7430   rankcrnk 7431
This theorem is referenced by:  rankidn  7490  rankpwi  7491  rankr1g  7500  r1tskina  8400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-rdg 6419  df-r1 7432  df-rank 7433
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