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Theorem rankr1ag 8124
Description: A version of rankr1a 8158 that is suitable without assuming Regularity or Replacement. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankr1ag  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  B
)  <->  ( rank `  A
)  e.  B ) )

Proof of Theorem rankr1ag
StepHypRef Expression
1 rankr1ai 8120 . 2  |-  ( A  e.  ( R1 `  B )  ->  ( rank `  A )  e.  B )
2 r1funlim 8088 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
32simpri 462 . . . . . . 7  |-  Lim  dom  R1
4 limord 4889 . . . . . . 7  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
53, 4ax-mp 5 . . . . . 6  |-  Ord  dom  R1
6 ordelord 4852 . . . . . 6  |-  ( ( Ord  dom  R1  /\  B  e.  dom  R1 )  ->  Ord  B )
75, 6mpan 670 . . . . 5  |-  ( B  e.  dom  R1  ->  Ord 
B )
87adantl 466 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  Ord  B )
9 ordsucss 6542 . . . 4  |-  ( Ord 
B  ->  ( ( rank `  A )  e.  B  ->  suc  ( rank `  A )  C_  B
) )
108, 9syl 16 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( rank `  A )  e.  B  ->  suc  ( rank `  A
)  C_  B )
)
11 rankidb 8122 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
12 elfvdm 5828 . . . . 5  |-  ( A  e.  ( R1 `  suc  ( rank `  A
) )  ->  suc  ( rank `  A )  e.  dom  R1 )
1311, 12syl 16 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  suc  ( rank `  A
)  e.  dom  R1 )
14 r1ord3g 8101 . . . 4  |-  ( ( suc  ( rank `  A
)  e.  dom  R1  /\  B  e.  dom  R1 )  ->  ( suc  ( rank `  A )  C_  B  ->  ( R1 `  suc  ( rank `  A
) )  C_  ( R1 `  B ) ) )
1513, 14sylan 471 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( suc  ( rank `  A )  C_  B  ->  ( R1 `  suc  ( rank `  A
) )  C_  ( R1 `  B ) ) )
1611adantr 465 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  A  e.  ( R1 `  suc  ( rank `  A ) ) )
17 ssel 3461 . . . 4  |-  ( ( R1 `  suc  ( rank `  A ) ) 
C_  ( R1 `  B )  ->  ( A  e.  ( R1 ` 
suc  ( rank `  A
) )  ->  A  e.  ( R1 `  B
) ) )
1816, 17syl5com 30 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( R1
`  suc  ( rank `  A ) )  C_  ( R1 `  B )  ->  A  e.  ( R1 `  B ) ) )
1910, 15, 183syld 55 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( rank `  A )  e.  B  ->  A  e.  ( R1
`  B ) ) )
201, 19impbid2 204 1  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  B
)  <->  ( rank `  A
)  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758    C_ wss 3439   U.cuni 4202   Ord word 4829   Oncon0 4830   Lim wlim 4831   suc csuc 4832   dom cdm 4951   "cima 4954   Fun wfun 5523   ` cfv 5529   R1cr1 8084   rankcrnk 8085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-recs 6945  df-rdg 6979  df-r1 8086  df-rank 8087
This theorem is referenced by:  rankr1bg  8125  rankr1clem  8142  rankr1c  8143  rankval3b  8148  onssr1  8153  r1pw  8167  r1pwcl  8169  hsmexlem6  8715  r1limwun  9018  inatsk  9060  grur1  9102
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