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Theorem rankr1ag 7728
Description: A version of rankr1a 7762 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 7724 . 2  |-  ( A  e.  ( R1 `  B )  ->  ( rank `  A )  e.  B )
2 r1funlim 7692 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
32simpri 449 . . . . . . 7  |-  Lim  dom  R1
4 limord 4640 . . . . . . 7  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
53, 4ax-mp 8 . . . . . 6  |-  Ord  dom  R1
6 ordelord 4603 . . . . . 6  |-  ( ( Ord  dom  R1  /\  B  e.  dom  R1 )  ->  Ord  B )
75, 6mpan 652 . . . . 5  |-  ( B  e.  dom  R1  ->  Ord 
B )
87adantl 453 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  Ord  B )
9 ordsucss 4798 . . . 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 7726 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
12 elfvdm 5757 . . . . 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 7705 . . . 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 458 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( suc  ( rank `  A )  C_  B  ->  ( R1 `  suc  ( rank `  A
) )  C_  ( R1 `  B ) ) )
1611adantr 452 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  A  e.  ( R1 `  suc  ( rank `  A ) ) )
17 ssel 3342 . . . 4  |-  ( ( R1 `  suc  ( rank `  A ) ) 
C_  ( R1 `  B )  ->  ( A  e.  ( R1 ` 
suc  ( rank `  A
) )  ->  A  e.  ( R1 `  B
) ) )
1816, 17syl5com 28 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( R1
`  suc  ( rank `  A ) )  C_  ( R1 `  B )  ->  A  e.  ( R1 `  B ) ) )
1910, 15, 183syld 53 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( rank `  A )  e.  B  ->  A  e.  ( R1
`  B ) ) )
201, 19impbid2 196 1  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  B
)  <->  ( rank `  A
)  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    C_ wss 3320   U.cuni 4015   Ord word 4580   Oncon0 4581   Lim wlim 4582   suc csuc 4583   dom cdm 4878   "cima 4881   Fun wfun 5448   ` cfv 5454   R1cr1 7688   rankcrnk 7689
This theorem is referenced by:  rankr1bg  7729  rankr1clem  7746  rankr1c  7747  rankval3b  7752  onssr1  7757  r1pw  7771  r1pwcl  7773  hsmexlem6  8311  r1limwun  8611  inatsk  8653  grur1  8695
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668  df-r1 7690  df-rank 7691
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