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Theorem uniwf 7748
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 7705 . . . . . . . 8  |-  Tr  ( R1 `  suc  ( rank `  A ) )
2 rankidb 7729 . . . . . . . 8  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
3 trss 4314 . . . . . . . 8  |-  ( Tr  ( R1 `  suc  ( rank `  A )
)  ->  ( A  e.  ( R1 `  suc  ( rank `  A )
)  ->  A  C_  ( R1 `  suc  ( rank `  A ) ) ) )
41, 2, 3mpsyl 62 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  suc  ( rank `  A
) ) )
5 rankdmr1 7730 . . . . . . . 8  |-  ( rank `  A )  e.  dom  R1
6 r1sucg 7698 . . . . . . . 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 3396 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ~P ( R1
`  ( rank `  A
) ) )
9 sspwuni 4179 . . . . . 6  |-  ( A 
C_  ~P ( R1 `  ( rank `  A )
)  <->  U. A  C_  ( R1 `  ( rank `  A
) ) )
108, 9sylib 190 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  U. A  C_  ( R1
`  ( rank `  A
) ) )
11 fvex 5745 . . . . . 6  |-  ( R1
`  ( rank `  A
) )  e.  _V
1211elpw2 4367 . . . . 5  |-  ( U. A  e.  ~P ( R1 `  ( rank `  A
) )  <->  U. A  C_  ( R1 `  ( rank `  A ) ) )
1310, 12sylibr 205 . . . 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 7725 . . 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 7736 . . 3  |-  ( U. A  e.  U. ( R1 " On )  <->  ~P U. A  e.  U. ( R1 " On ) )
18 pwuni 4398 . . . 4  |-  A  C_  ~P U. A
19 sswf 7737 . . . 4  |-  ( ( ~P U. A  e. 
U. ( R1 " On )  /\  A  C_  ~P U. A )  ->  A  e.  U. ( R1 " On ) )
2018, 19mpan2 654 . . 3  |-  ( ~P
U. A  e.  U. ( R1 " On )  ->  A  e.  U. ( R1 " On ) )
2117, 20sylbi 189 . 2  |-  ( U. A  e.  U. ( R1 " On )  ->  A  e.  U. ( R1 " On ) )
2216, 21impbii 182 1  |-  ( A  e.  U. ( R1
" On )  <->  U. A  e. 
U. ( R1 " On ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   Tr wtr 4305   Oncon0 4584   suc csuc 4586   dom cdm 4881   "cima 4884   ` cfv 5457   R1cr1 7691   rankcrnk 7692
This theorem is referenced by:  rankuni2b  7782  r1limwun  8616  wfgru  8696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-recs 6636  df-rdg 6671  df-r1 7693  df-rank 7694
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