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Theorem uniwf 8308
Description: A union is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
uniwf  |-  ( A  e.  U. ( R1
" On )  <->  U. A  e. 
U. ( R1 " On ) )

Proof of Theorem uniwf
StepHypRef Expression
1 r1tr 8265 . . . . . . . 8  |-  Tr  ( R1 `  suc  ( rank `  A ) )
2 rankidb 8289 . . . . . . . 8  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
3 trss 4499 . . . . . . . 8  |-  ( Tr  ( R1 `  suc  ( rank `  A )
)  ->  ( A  e.  ( R1 `  suc  ( rank `  A )
)  ->  A  C_  ( R1 `  suc  ( rank `  A ) ) ) )
41, 2, 3mpsyl 64 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  suc  ( rank `  A
) ) )
5 rankdmr1 8290 . . . . . . . 8  |-  ( rank `  A )  e.  dom  R1
6 r1sucg 8258 . . . . . . . 8  |-  ( (
rank `  A )  e.  dom  R1  ->  ( R1 `  suc  ( rank `  A ) )  =  ~P ( R1 `  ( rank `  A )
) )
75, 6ax-mp 5 . . . . . . 7  |-  ( R1
`  suc  ( rank `  A ) )  =  ~P ( R1 `  ( rank `  A )
)
84, 7syl6sseq 3464 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ~P ( R1
`  ( rank `  A
) ) )
9 sspwuni 4360 . . . . . 6  |-  ( A 
C_  ~P ( R1 `  ( rank `  A )
)  <->  U. A  C_  ( R1 `  ( rank `  A
) ) )
108, 9sylib 201 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  U. A  C_  ( R1
`  ( rank `  A
) ) )
11 fvex 5889 . . . . . 6  |-  ( R1
`  ( rank `  A
) )  e.  _V
1211elpw2 4565 . . . . 5  |-  ( U. A  e.  ~P ( R1 `  ( rank `  A
) )  <->  U. A  C_  ( R1 `  ( rank `  A ) ) )
1310, 12sylibr 217 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  U. A  e.  ~P ( R1 `  ( rank `  A ) ) )
1413, 7syl6eleqr 2560 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  U. A  e.  ( R1 `  suc  ( rank `  A ) ) )
15 r1elwf 8285 . . 3  |-  ( U. A  e.  ( R1 ` 
suc  ( rank `  A
) )  ->  U. A  e.  U. ( R1 " On ) )
1614, 15syl 17 . 2  |-  ( A  e.  U. ( R1
" On )  ->  U. A  e.  U. ( R1 " On ) )
17 pwwf 8296 . . 3  |-  ( U. A  e.  U. ( R1 " On )  <->  ~P U. A  e.  U. ( R1 " On ) )
18 pwuni 4631 . . . 4  |-  A  C_  ~P U. A
19 sswf 8297 . . . 4  |-  ( ( ~P U. A  e. 
U. ( R1 " On )  /\  A  C_  ~P U. A )  ->  A  e.  U. ( R1 " On ) )
2018, 19mpan2 685 . . 3  |-  ( ~P
U. A  e.  U. ( R1 " On )  ->  A  e.  U. ( R1 " On ) )
2117, 20sylbi 200 . 2  |-  ( U. A  e.  U. ( R1 " On )  ->  A  e.  U. ( R1 " On ) )
2216, 21impbii 192 1  |-  ( A  e.  U. ( R1
" On )  <->  U. A  e. 
U. ( R1 " On ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    = wceq 1452    e. wcel 1904    C_ wss 3390   ~Pcpw 3942   U.cuni 4190   Tr wtr 4490   dom cdm 4839   "cima 4842   Oncon0 5430   suc csuc 5432   ` cfv 5589   R1cr1 8251   rankcrnk 8252
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-int 4227  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  df-rank 8254
This theorem is referenced by:  rankuni2b  8342  r1limwun  9179  wfgru  9259
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