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Theorem tz9.12 7457
Description: A set is well-founded if all of its elements are well-founded. Proposition 9.12 of [TakeutiZaring] p. 78. The main proof consists of tz9.12lem1 7454 through tz9.12lem3 7456. (Contributed by NM, 22-Sep-2003.)
Hypothesis
Ref Expression
tz9.12.1  |-  A  e. 
_V
Assertion
Ref Expression
tz9.12  |-  ( A. x  e.  A  E. y  e.  On  x  e.  ( R1 `  y
)  ->  E. y  e.  On  A  e.  ( R1 `  y ) )
Distinct variable group:    x, y, A
Dummy variables  z 
v are mutually distinct and distinct from all other variables.

Proof of Theorem tz9.12
StepHypRef Expression
1 tz9.12.1 . . . 4  |-  A  e. 
_V
2 eqid 2284 . . . 4  |-  ( z  e.  _V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v ) } )  =  ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } )
31, 2tz9.12lem2 7455 . . 3  |-  suc  U. ( ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } ) " A )  e.  On
43onsuci 4628 . 2  |-  suc  suc  U. ( ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } ) " A )  e.  On
51, 2tz9.12lem3 7456 . 2  |-  ( A. x  e.  A  E. y  e.  On  x  e.  ( R1 `  y
)  ->  A  e.  ( R1 `  suc  suc  U. ( ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } ) " A ) ) )
6 fveq2 5485 . . . 4  |-  ( y  =  suc  suc  U. ( ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } ) " A )  ->  ( R1 `  y )  =  ( R1 `  suc  suc  U. ( ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } ) " A ) ) )
76eleq2d 2351 . . 3  |-  ( y  =  suc  suc  U. ( ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } ) " A )  ->  ( A  e.  ( R1 `  y )  <->  A  e.  ( R1 `  suc  suc  U. ( ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } ) " A ) ) ) )
87rspcev 2885 . 2  |-  ( ( suc  suc  U. (
( z  e.  _V  |->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) } ) " A )  e.  On  /\  A  e.  ( R1 `  suc  suc  U. ( ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } ) " A ) ) )  ->  E. y  e.  On  A  e.  ( R1 `  y ) )
94, 5, 8sylancr 646 1  |-  ( A. x  e.  A  E. y  e.  On  x  e.  ( R1 `  y
)  ->  E. y  e.  On  A  e.  ( R1 `  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   A.wral 2544   E.wrex 2545   {crab 2548   _Vcvv 2789   U.cuni 3828   |^|cint 3863    e. cmpt 4078   Oncon0 4391   suc csuc 4393   "cima 4691   ` cfv 5221   R1cr1 7429
This theorem is referenced by:  tz9.13  7458  r1elss  7473
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 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6383  df-rdg 6418  df-r1 7431
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