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Theorem r1elwf 8285
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 8255 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 469 . . . . 5  |-  Lim  dom  R1
3 limord 5489 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 6635 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 10 . . . 4  |-  dom  R1  C_  On
6 elfvdm 5905 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  B  e.  dom  R1 )
75, 6sseldi 3416 . . 3  |-  ( A  e.  ( R1 `  B )  ->  B  e.  On )
8 r1tr 8265 . . . . . 6  |-  Tr  ( R1 `  B )
9 trss 4499 . . . . . 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 3950 . . . . 5  |-  ( A  e.  ( R1 `  B )  ->  ( A  e.  ~P ( R1 `  B )  <->  A  C_  ( R1 `  B ) ) )
1210, 11mpbird 240 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  ~P ( R1 `  B ) )
13 r1sucg 8258 . . . . 5  |-  ( B  e.  dom  R1  ->  ( R1 `  suc  B
)  =  ~P ( R1 `  B ) )
146, 13syl 17 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  ( R1 `  suc  B )  =  ~P ( R1
`  B ) )
1512, 14eleqtrrd 2552 . . 3  |-  ( A  e.  ( R1 `  B )  ->  A  e.  ( R1 `  suc  B ) )
16 suceq 5495 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
1716fveq2d 5883 . . . . 5  |-  ( x  =  B  ->  ( R1 `  suc  x )  =  ( R1 `  suc  B ) )
1817eleq2d 2534 . . . 4  |-  ( x  =  B  ->  ( A  e.  ( R1 ` 
suc  x )  <->  A  e.  ( R1 `  suc  B
) ) )
1918rspcev 3136 . . 3  |-  ( ( B  e.  On  /\  A  e.  ( R1 ` 
suc  B ) )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
207, 15, 19syl2anc 673 . 2  |-  ( A  e.  ( R1 `  B )  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
21 rankwflemb 8282 . 2  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
2220, 21sylibr 217 1  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   E.wrex 2757    C_ wss 3390   ~Pcpw 3942   U.cuni 4190   Tr wtr 4490   dom cdm 4839   "cima 4842   Ord word 5429   Oncon0 5430   Lim wlim 5431   suc csuc 5432   Fun wfun 5583   ` cfv 5589   R1cr1 8251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-r1 8253
This theorem is referenced by:  rankr1ai  8287  pwwf  8296  sswf  8297  unwf  8299  uniwf  8308  rankonidlem  8317  r1pw  8334  r1pwcl  8336  rankr1id  8351  tcrank  8373  dfac12lem2  8592  r1limwun  9179  r1wunlim  9180  inatsk  9221
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