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Theorem r1elwf 7999
Description: Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1elwf  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )

Proof of Theorem r1elwf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 r1funlim 7969 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 459 . . . . 5  |-  Lim  dom  R1
3 limord 4774 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 6400 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 10 . . . 4  |-  dom  R1  C_  On
6 elfvdm 5713 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  B  e.  dom  R1 )
75, 6sseldi 3351 . . 3  |-  ( A  e.  ( R1 `  B )  ->  B  e.  On )
8 r1tr 7979 . . . . . 6  |-  Tr  ( R1 `  B )
9 trss 4391 . . . . . 6  |-  ( Tr  ( R1 `  B
)  ->  ( A  e.  ( R1 `  B
)  ->  A  C_  ( R1 `  B ) ) )
108, 9ax-mp 5 . . . . 5  |-  ( A  e.  ( R1 `  B )  ->  A  C_  ( R1 `  B
) )
11 elpwg 3865 . . . . 5  |-  ( A  e.  ( R1 `  B )  ->  ( A  e.  ~P ( R1 `  B )  <->  A  C_  ( R1 `  B ) ) )
1210, 11mpbird 232 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  ~P ( R1 `  B ) )
13 r1sucg 7972 . . . . 5  |-  ( B  e.  dom  R1  ->  ( R1 `  suc  B
)  =  ~P ( R1 `  B ) )
146, 13syl 16 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  ( R1 `  suc  B )  =  ~P ( R1
`  B ) )
1512, 14eleqtrrd 2518 . . 3  |-  ( A  e.  ( R1 `  B )  ->  A  e.  ( R1 `  suc  B ) )
16 suceq 4780 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
1716fveq2d 5692 . . . . 5  |-  ( x  =  B  ->  ( R1 `  suc  x )  =  ( R1 `  suc  B ) )
1817eleq2d 2508 . . . 4  |-  ( x  =  B  ->  ( A  e.  ( R1 ` 
suc  x )  <->  A  e.  ( R1 `  suc  B
) ) )
1918rspcev 3070 . . 3  |-  ( ( B  e.  On  /\  A  e.  ( R1 ` 
suc  B ) )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
207, 15, 19syl2anc 656 . 2  |-  ( A  e.  ( R1 `  B )  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
21 rankwflemb 7996 . 2  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
2220, 21sylibr 212 1  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761   E.wrex 2714    C_ wss 3325   ~Pcpw 3857   U.cuni 4088   Tr wtr 4382   Ord word 4714   Oncon0 4715   Lim wlim 4716   suc csuc 4717   dom cdm 4836   "cima 4839   Fun wfun 5409   ` cfv 5415   R1cr1 7965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-recs 6828  df-rdg 6862  df-r1 7967
This theorem is referenced by:  rankr1ai  8001  pwwf  8010  sswf  8011  unwf  8013  uniwf  8022  rankonidlem  8031  r1pw  8048  r1pwcl  8050  rankr1id  8065  tcrank  8087  dfac12lem2  8309  r1limwun  8899  r1wunlim  8900  inatsk  8941
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