MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankr1ag Structured version   Unicode version

Theorem rankr1ag 8225
Description: A version of rankr1a 8259 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 8221 . 2  |-  ( A  e.  ( R1 `  B )  ->  ( rank `  A )  e.  B )
2 r1funlim 8189 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
32simpri 463 . . . . . . 7  |-  Lim  dom  R1
4 limord 5444 . . . . . . 7  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
53, 4ax-mp 5 . . . . . 6  |-  Ord  dom  R1
6 ordelord 5407 . . . . . 6  |-  ( ( Ord  dom  R1  /\  B  e.  dom  R1 )  ->  Ord  B )
75, 6mpan 674 . . . . 5  |-  ( B  e.  dom  R1  ->  Ord 
B )
87adantl 467 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  Ord  B )
9 ordsucss 6603 . . . 4  |-  ( Ord 
B  ->  ( ( rank `  A )  e.  B  ->  suc  ( rank `  A )  C_  B
) )
108, 9syl 17 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( rank `  A )  e.  B  ->  suc  ( rank `  A
)  C_  B )
)
11 rankidb 8223 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
12 elfvdm 5851 . . . . 5  |-  ( A  e.  ( R1 `  suc  ( rank `  A
) )  ->  suc  ( rank `  A )  e.  dom  R1 )
1311, 12syl 17 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  suc  ( rank `  A
)  e.  dom  R1 )
14 r1ord3g 8202 . . . 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 473 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( suc  ( rank `  A )  C_  B  ->  ( R1 `  suc  ( rank `  A
) )  C_  ( R1 `  B ) ) )
1611adantr 466 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  A  e.  ( R1 `  suc  ( rank `  A ) ) )
17 ssel 3401 . . . 4  |-  ( ( R1 `  suc  ( rank `  A ) ) 
C_  ( R1 `  B )  ->  ( A  e.  ( R1 ` 
suc  ( rank `  A
) )  ->  A  e.  ( R1 `  B
) ) )
1816, 17syl5com 31 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( R1
`  suc  ( rank `  A ) )  C_  ( R1 `  B )  ->  A  e.  ( R1 `  B ) ) )
1910, 15, 183syld 57 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( rank `  A )  e.  B  ->  A  e.  ( R1
`  B ) ) )
201, 19impbid2 207 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 187    /\ wa 370    e. wcel 1872    C_ wss 3379   U.cuni 4162   dom cdm 4796   "cima 4799   Ord word 5384   Oncon0 5385   Lim wlim 5386   suc csuc 5387   Fun wfun 5538   ` cfv 5544   R1cr1 8185   rankcrnk 8186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-r1 8187  df-rank 8188
This theorem is referenced by:  rankr1bg  8226  rankr1clem  8243  rankr1c  8244  rankval3b  8249  onssr1  8254  r1pw  8268  r1pwcl  8270  hsmexlem6  8812  r1limwun  9112  inatsk  9154  grur1  9196
  Copyright terms: Public domain W3C validator