MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1elwf Structured version   Unicode version

Theorem r1elwf 8205
Description: Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1elwf  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )

Proof of Theorem r1elwf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 r1funlim 8175 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 460 . . . . 5  |-  Lim  dom  R1
3 limord 4926 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 6598 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 10 . . . 4  |-  dom  R1  C_  On
6 elfvdm 5874 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  B  e.  dom  R1 )
75, 6sseldi 3487 . . 3  |-  ( A  e.  ( R1 `  B )  ->  B  e.  On )
8 r1tr 8185 . . . . . 6  |-  Tr  ( R1 `  B )
9 trss 4541 . . . . . 6  |-  ( Tr  ( R1 `  B
)  ->  ( A  e.  ( R1 `  B
)  ->  A  C_  ( R1 `  B ) ) )
108, 9ax-mp 5 . . . . 5  |-  ( A  e.  ( R1 `  B )  ->  A  C_  ( R1 `  B
) )
11 elpwg 4007 . . . . 5  |-  ( A  e.  ( R1 `  B )  ->  ( A  e.  ~P ( R1 `  B )  <->  A  C_  ( R1 `  B ) ) )
1210, 11mpbird 232 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  A  e.  ~P ( R1 `  B ) )
13 r1sucg 8178 . . . . 5  |-  ( B  e.  dom  R1  ->  ( R1 `  suc  B
)  =  ~P ( R1 `  B ) )
146, 13syl 16 . . . 4  |-  ( A  e.  ( R1 `  B )  ->  ( R1 `  suc  B )  =  ~P ( R1
`  B ) )
1512, 14eleqtrrd 2545 . . 3  |-  ( A  e.  ( R1 `  B )  ->  A  e.  ( R1 `  suc  B ) )
16 suceq 4932 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
1716fveq2d 5852 . . . . 5  |-  ( x  =  B  ->  ( R1 `  suc  x )  =  ( R1 `  suc  B ) )
1817eleq2d 2524 . . . 4  |-  ( x  =  B  ->  ( A  e.  ( R1 ` 
suc  x )  <->  A  e.  ( R1 `  suc  B
) ) )
1918rspcev 3207 . . 3  |-  ( ( B  e.  On  /\  A  e.  ( R1 ` 
suc  B ) )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
207, 15, 19syl2anc 659 . 2  |-  ( A  e.  ( R1 `  B )  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
21 rankwflemb 8202 . 2  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
2220, 21sylibr 212 1  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   E.wrex 2805    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   Tr wtr 4532   Ord word 4866   Oncon0 4867   Lim wlim 4868   suc csuc 4869   dom cdm 4988   "cima 4991   Fun wfun 5564   ` cfv 5570   R1cr1 8171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173
This theorem is referenced by:  rankr1ai  8207  pwwf  8216  sswf  8217  unwf  8219  uniwf  8228  rankonidlem  8237  r1pw  8254  r1pwcl  8256  rankr1id  8271  tcrank  8293  dfac12lem2  8515  r1limwun  9103  r1wunlim  9104  inatsk  9145
  Copyright terms: Public domain W3C validator