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Theorem tz9.12 7705
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 7702 through tz9.12lem3 7704. (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

Proof of Theorem tz9.12
Dummy variables  z 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tz9.12.1 . . . 4  |-  A  e. 
_V
2 eqid 2435 . . . 4  |-  ( z  e.  _V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v ) } )  =  ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } )
31, 2tz9.12lem2 7703 . . 3  |-  suc  U. ( ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } ) " A )  e.  On
43onsuci 4809 . 2  |-  suc  suc  U. ( ( z  e. 
_V  |->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } ) " A )  e.  On
51, 2tz9.12lem3 7704 . 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 5719 . . . 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 2502 . . 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 3044 . 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 645 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 4    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948   U.cuni 4007   |^|cint 4042    e. cmpt 4258   Oncon0 4573   suc csuc 4575   "cima 4872   ` cfv 5445   R1cr1 7677
This theorem is referenced by:  tz9.13  7706  r1elss  7721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-recs 6624  df-rdg 6659  df-r1 7679
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