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Theorem uniwf 8014
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 7971 . . . . . . . 8  |-  Tr  ( R1 `  suc  ( rank `  A ) )
2 rankidb 7995 . . . . . . . 8  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
3 trss 4382 . . . . . . . 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 7996 . . . . . . . 8  |-  ( rank `  A )  e.  dom  R1
6 r1sucg 7964 . . . . . . . 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 3390 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ~P ( R1
`  ( rank `  A
) ) )
9 sspwuni 4244 . . . . . 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 5689 . . . . . 6  |-  ( R1
`  ( rank `  A
) )  e.  _V
1211elpw2 4444 . . . . 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 2524 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  U. A  e.  ( R1 `  suc  ( rank `  A ) ) )
15 r1elwf 7991 . . 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 8002 . . 3  |-  ( U. A  e.  U. ( R1 " On )  <->  ~P U. A  e.  U. ( R1 " On ) )
18 pwuni 4511 . . . 4  |-  A  C_  ~P U. A
19 sswf 8003 . . . 4  |-  ( ( ~P U. A  e. 
U. ( R1 " On )  /\  A  C_  ~P U. A )  ->  A  e.  U. ( R1 " On ) )
2018, 19mpan2 664 . . 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 1362    e. wcel 1755    C_ wss 3316   ~Pcpw 3848   U.cuni 4079   Tr wtr 4373   Oncon0 4706   suc csuc 4708   dom cdm 4827   "cima 4830   ` cfv 5406   R1cr1 7957   rankcrnk 7958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-om 6466  df-recs 6818  df-rdg 6852  df-r1 7959  df-rank 7960
This theorem is referenced by:  rankuni2b  8048  r1limwun  8891  wfgru  8971
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