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Theorem uniwf 8018
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 7975 . . . . . . . 8  |-  Tr  ( R1 `  suc  ( rank `  A ) )
2 rankidb 7999 . . . . . . . 8  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
3 trss 4389 . . . . . . . 8  |-  ( Tr  ( R1 `  suc  ( rank `  A )
)  ->  ( A  e.  ( R1 `  suc  ( rank `  A )
)  ->  A  C_  ( R1 `  suc  ( rank `  A ) ) ) )
41, 2, 3mpsyl 63 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  suc  ( rank `  A
) ) )
5 rankdmr1 8000 . . . . . . . 8  |-  ( rank `  A )  e.  dom  R1
6 r1sucg 7968 . . . . . . . 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 3397 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ~P ( R1
`  ( rank `  A
) ) )
9 sspwuni 4251 . . . . . 6  |-  ( A 
C_  ~P ( R1 `  ( rank `  A )
)  <->  U. A  C_  ( R1 `  ( rank `  A
) ) )
108, 9sylib 196 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  U. A  C_  ( R1
`  ( rank `  A
) ) )
11 fvex 5696 . . . . . 6  |-  ( R1
`  ( rank `  A
) )  e.  _V
1211elpw2 4451 . . . . 5  |-  ( U. A  e.  ~P ( R1 `  ( rank `  A
) )  <->  U. A  C_  ( R1 `  ( rank `  A ) ) )
1310, 12sylibr 212 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  U. A  e.  ~P ( R1 `  ( rank `  A ) ) )
1413, 7syl6eleqr 2529 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  U. A  e.  ( R1 `  suc  ( rank `  A ) ) )
15 r1elwf 7995 . . 3  |-  ( U. A  e.  ( R1 ` 
suc  ( rank `  A
) )  ->  U. A  e.  U. ( R1 " On ) )
1614, 15syl 16 . 2  |-  ( A  e.  U. ( R1
" On )  ->  U. A  e.  U. ( R1 " On ) )
17 pwwf 8006 . . 3  |-  ( U. A  e.  U. ( R1 " On )  <->  ~P U. A  e.  U. ( R1 " On ) )
18 pwuni 4518 . . . 4  |-  A  C_  ~P U. A
19 sswf 8007 . . . 4  |-  ( ( ~P U. A  e. 
U. ( R1 " On )  /\  A  C_  ~P U. A )  ->  A  e.  U. ( R1 " On ) )
2018, 19mpan2 671 . . 3  |-  ( ~P
U. A  e.  U. ( R1 " On )  ->  A  e.  U. ( R1 " On ) )
2117, 20sylbi 195 . 2  |-  ( U. A  e.  U. ( R1 " On )  ->  A  e.  U. ( R1 " On ) )
2216, 21impbii 188 1  |-  ( A  e.  U. ( R1
" On )  <->  U. A  e. 
U. ( R1 " On ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    e. wcel 1756    C_ wss 3323   ~Pcpw 3855   U.cuni 4086   Tr wtr 4380   Oncon0 4714   suc csuc 4716   dom cdm 4835   "cima 4838   ` cfv 5413   R1cr1 7961   rankcrnk 7962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-recs 6824  df-rdg 6858  df-r1 7963  df-rank 7964
This theorem is referenced by:  rankuni2b  8052  r1limwun  8895  wfgru  8975
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