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Theorem r1elwf 8212
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 8182 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 462 . . . . 5  |-  Lim  dom  R1
3 limord 4923 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 6606 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 10 . . . 4  |-  dom  R1  C_  On
6 elfvdm 5878 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  B  e.  dom  R1 )
75, 6sseldi 3484 . . 3  |-  ( A  e.  ( R1 `  B )  ->  B  e.  On )
8 r1tr 8192 . . . . . 6  |-  Tr  ( R1 `  B )
9 trss 4535 . . . . . 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 4001 . . . . 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 8185 . . . . 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 2532 . . 3  |-  ( A  e.  ( R1 `  B )  ->  A  e.  ( R1 `  suc  B ) )
16 suceq 4929 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
1716fveq2d 5856 . . . . 5  |-  ( x  =  B  ->  ( R1 `  suc  x )  =  ( R1 `  suc  B ) )
1817eleq2d 2511 . . . 4  |-  ( x  =  B  ->  ( A  e.  ( R1 ` 
suc  x )  <->  A  e.  ( R1 `  suc  B
) ) )
1918rspcev 3194 . . 3  |-  ( ( B  e.  On  /\  A  e.  ( R1 ` 
suc  B ) )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
207, 15, 19syl2anc 661 . 2  |-  ( A  e.  ( R1 `  B )  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
21 rankwflemb 8209 . 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 1381    e. wcel 1802   E.wrex 2792    C_ wss 3458   ~Pcpw 3993   U.cuni 4230   Tr wtr 4526   Ord word 4863   Oncon0 4864   Lim wlim 4865   suc csuc 4866   dom cdm 4985   "cima 4988   Fun wfun 5568   ` cfv 5574   R1cr1 8178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-om 6682  df-recs 7040  df-rdg 7074  df-r1 8180
This theorem is referenced by:  rankr1ai  8214  pwwf  8223  sswf  8224  unwf  8226  uniwf  8235  rankonidlem  8244  r1pw  8261  r1pwcl  8263  rankr1id  8278  tcrank  8300  dfac12lem2  8522  r1limwun  9112  r1wunlim  9113  inatsk  9154
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