MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniwf Unicode version

Theorem uniwf 7581
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 7538 . . . . . . . 8  |-  Tr  ( R1 `  suc  ( rank `  A ) )
2 rankidb 7562 . . . . . . . 8  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
3 trss 4203 . . . . . . . 8  |-  ( Tr  ( R1 `  suc  ( rank `  A )
)  ->  ( A  e.  ( R1 `  suc  ( rank `  A )
)  ->  A  C_  ( R1 `  suc  ( rank `  A ) ) ) )
41, 2, 3mpsyl 59 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ( R1 `  suc  ( rank `  A
) ) )
5 rankdmr1 7563 . . . . . . . 8  |-  ( rank `  A )  e.  dom  R1
6 r1sucg 7531 . . . . . . . 8  |-  ( (
rank `  A )  e.  dom  R1  ->  ( R1 `  suc  ( rank `  A ) )  =  ~P ( R1 `  ( rank `  A )
) )
75, 6ax-mp 8 . . . . . . 7  |-  ( R1
`  suc  ( rank `  A ) )  =  ~P ( R1 `  ( rank `  A )
)
84, 7syl6sseq 3300 . . . . . 6  |-  ( A  e.  U. ( R1
" On )  ->  A  C_  ~P ( R1
`  ( rank `  A
) ) )
9 sspwuni 4068 . . . . . 6  |-  ( A 
C_  ~P ( R1 `  ( rank `  A )
)  <->  U. A  C_  ( R1 `  ( rank `  A
) ) )
108, 9sylib 188 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  U. A  C_  ( R1
`  ( rank `  A
) ) )
11 fvex 5622 . . . . . 6  |-  ( R1
`  ( rank `  A
) )  e.  _V
1211elpw2 4256 . . . . 5  |-  ( U. A  e.  ~P ( R1 `  ( rank `  A
) )  <->  U. A  C_  ( R1 `  ( rank `  A ) ) )
1310, 12sylibr 203 . . . 4  |-  ( A  e.  U. ( R1
" On )  ->  U. A  e.  ~P ( R1 `  ( rank `  A ) ) )
1413, 7syl6eleqr 2449 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  U. A  e.  ( R1 `  suc  ( rank `  A ) ) )
15 r1elwf 7558 . . 3  |-  ( U. A  e.  ( R1 ` 
suc  ( rank `  A
) )  ->  U. A  e.  U. ( R1 " On ) )
1614, 15syl 15 . 2  |-  ( A  e.  U. ( R1
" On )  ->  U. A  e.  U. ( R1 " On ) )
17 pwwf 7569 . . 3  |-  ( U. A  e.  U. ( R1 " On )  <->  ~P U. A  e.  U. ( R1 " On ) )
18 pwuni 4287 . . . 4  |-  A  C_  ~P U. A
19 sswf 7570 . . . 4  |-  ( ( ~P U. A  e. 
U. ( R1 " On )  /\  A  C_  ~P U. A )  ->  A  e.  U. ( R1 " On ) )
2018, 19mpan2 652 . . 3  |-  ( ~P
U. A  e.  U. ( R1 " On )  ->  A  e.  U. ( R1 " On ) )
2117, 20sylbi 187 . 2  |-  ( U. A  e.  U. ( R1 " On )  ->  A  e.  U. ( R1 " On ) )
2216, 21impbii 180 1  |-  ( A  e.  U. ( R1
" On )  <->  U. A  e. 
U. ( R1 " On ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1642    e. wcel 1710    C_ wss 3228   ~Pcpw 3701   U.cuni 3908   Tr wtr 4194   Oncon0 4474   suc csuc 4476   dom cdm 4771   "cima 4774   ` cfv 5337   R1cr1 7524   rankcrnk 7525
This theorem is referenced by:  rankuni2b  7615  r1limwun  8448  wfgru  8528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-recs 6475  df-rdg 6510  df-r1 7526  df-rank 7527
  Copyright terms: Public domain W3C validator