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Theorem rankwflem 7483
Description: Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 7460 is useful in proofs of theorems about the rank function. (Contributed by NM, 4-Oct-2003.)
Assertion
Ref Expression
rankwflem  |-  ( A  e.  V  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem rankwflem
StepHypRef Expression
1 elex 2798 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
2 unir1 7481 . . 3  |-  U. ( R1 " On )  =  _V
31, 2syl6eleqr 2376 . 2  |-  ( A  e.  V  ->  A  e.  U. ( R1 " On ) )
4 rankwflemb 7461 . 2  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
53, 4sylib 190 1  |-  ( A  e.  V  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1685   E.wrex 2546   _Vcvv 2790   U.cuni 3829   Oncon0 4392   suc csuc 4394   "cima 4692   ` cfv 5222   R1cr1 7430
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7302  ax-inf2 7338
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-recs 6384  df-rdg 6419  df-r1 7432
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