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Theorem rankr1c 7739
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 20 . . . 4  |-  ( B  =  ( rank `  A
)  ->  B  =  ( rank `  A )
)
2 rankdmr1 7719 . . . 4  |-  ( rank `  A )  e.  dom  R1
31, 2syl6eqel 2523 . . 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 5749 . . . . 5  |-  ( A  e.  ( R1 `  suc  B )  ->  suc  B  e.  dom  R1 )
6 r1funlim 7684 . . . . . . 7  |-  ( Fun 
R1  /\  Lim  dom  R1 )
76simpri 449 . . . . . 6  |-  Lim  dom  R1
8 limsuc 4821 . . . . . 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 204 . . . 4  |-  ( A  e.  ( R1 `  suc  B )  ->  B  e.  dom  R1 )
1110adantl 453 . . 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 7738 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( -.  A  e.  ( R1 `  B
)  <->  B  C_  ( rank `  A ) ) )
14 rankr1ag 7720 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  suc  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
159, 14sylan2b 462 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
16 rankon 7713 . . . . . . 7  |-  ( rank `  A )  e.  On
17 limord 4632 . . . . . . . . . 10  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
187, 17ax-mp 8 . . . . . . . . 9  |-  Ord  dom  R1
19 ordelon 4597 . . . . . . . . 9  |-  ( ( Ord  dom  R1  /\  B  e.  dom  R1 )  ->  B  e.  On )
2018, 19mpan 652 . . . . . . . 8  |-  ( B  e.  dom  R1  ->  B  e.  On )
2120adantl 453 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  B  e.  On )
22 onsssuc 4661 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  On  /\  B  e.  On )  ->  ( ( rank `  A
)  C_  B  <->  ( rank `  A )  e.  suc  B ) )
2316, 21, 22sylancr 645 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( rank `  A )  C_  B  <->  (
rank `  A )  e.  suc  B ) )
2415, 23bitr4d 248 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  C_  B )
)
2513, 24anbi12d 692 . . . 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 3355 . . . 4  |-  ( B  =  ( rank `  A
)  <->  ( B  C_  ( rank `  A )  /\  ( rank `  A
)  C_  B )
)
2725, 26syl6rbbr 256 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( B  =  ( rank `  A
)  <->  ( -.  A  e.  ( R1 `  B
)  /\  A  e.  ( R1 `  suc  B
) ) ) )
2827ex 424 . 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 344 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   U.cuni 4007   Ord word 4572   Oncon0 4573   Lim wlim 4574   suc csuc 4575   dom cdm 4870   "cima 4873   Fun wfun 5440   ` cfv 5446   R1cr1 7680   rankcrnk 7681
This theorem is referenced by:  rankidn  7740  rankpwi  7741  rankr1g  7750  r1tskina  8649
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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-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-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-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-recs 6625  df-rdg 6660  df-r1 7682  df-rank 7683
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