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Theorem rankr1c 7461
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 21 . . . 4  |-  ( B  =  ( rank `  A
)  ->  B  =  ( rank `  A )
)
2 rankdmr1 7441 . . . 4  |-  ( rank `  A )  e.  dom  R1
31, 2syl6eqel 2346 . . 3  |-  ( B  =  ( rank `  A
)  ->  B  e.  dom  R1 )
43a1i 12 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( B  =  (
rank `  A )  ->  B  e.  dom  R1 ) )
5 elfvdm 5488 . . . . 5  |-  ( A  e.  ( R1 `  suc  B )  ->  suc  B  e.  dom  R1 )
6 r1funlim 7406 . . . . . . 7  |-  ( Fun 
R1  /\  Lim  dom  R1 )
76simpri 450 . . . . . 6  |-  Lim  dom  R1
8 limsuc 4612 . . . . . 6  |-  ( Lim 
dom  R1  ->  ( B  e.  dom  R1  <->  suc  B  e. 
dom  R1 ) )
97, 8ax-mp 10 . . . . 5  |-  ( B  e.  dom  R1  <->  suc  B  e. 
dom  R1 )
105, 9sylibr 205 . . . 4  |-  ( A  e.  ( R1 `  suc  B )  ->  B  e.  dom  R1 )
1110adantl 454 . . 3  |-  ( ( -.  A  e.  ( R1 `  B )  /\  A  e.  ( R1 `  suc  B
) )  ->  B  e.  dom  R1 )
1211a1i 12 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( ( -.  A  e.  ( R1 `  B
)  /\  A  e.  ( R1 `  suc  B
) )  ->  B  e.  dom  R1 ) )
13 rankr1clem 7460 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( -.  A  e.  ( R1 `  B
)  <->  B  C_  ( rank `  A ) ) )
14 rankr1ag 7442 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  suc  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
159, 14sylan2b 463 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
16 rankon 7435 . . . . . . 7  |-  ( rank `  A )  e.  On
17 limord 4423 . . . . . . . . . 10  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
187, 17ax-mp 10 . . . . . . . . 9  |-  Ord  dom  R1
19 ordelon 4388 . . . . . . . . 9  |-  ( ( Ord  dom  R1  /\  B  e.  dom  R1 )  ->  B  e.  On )
2018, 19mpan 654 . . . . . . . 8  |-  ( B  e.  dom  R1  ->  B  e.  On )
2120adantl 454 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  B  e.  On )
22 onsssuc 4452 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  On  /\  B  e.  On )  ->  ( ( rank `  A
)  C_  B  <->  ( rank `  A )  e.  suc  B ) )
2316, 21, 22sylancr 647 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( rank `  A )  C_  B  <->  (
rank `  A )  e.  suc  B ) )
2415, 23bitr4d 249 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  C_  B )
)
2513, 24anbi12d 694 . . . 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 3169 . . . 4  |-  ( B  =  ( rank `  A
)  <->  ( B  C_  ( rank `  A )  /\  ( rank `  A
)  C_  B )
)
2725, 26syl6rbbr 257 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( B  =  ( rank `  A
)  <->  ( -.  A  e.  ( R1 `  B
)  /\  A  e.  ( R1 `  suc  B
) ) ) )
2827ex 425 . 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 345 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 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    C_ wss 3127   U.cuni 3801   Ord word 4363   Oncon0 4364   Lim wlim 4365   suc csuc 4366   dom cdm 4661   "cima 4664   Fun wfun 4667   ` cfv 4673   R1cr1 7402   rankcrnk 7403
This theorem is referenced by:  rankidn  7462  rankpwi  7463  rankr1g  7472  r1tskina  8372
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-recs 6356  df-rdg 6391  df-r1 7404  df-rank 7405
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