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Theorem tz9.13 7706
Description: Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78. (Contributed by NM, 23-Sep-2003.)
Hypothesis
Ref Expression
tz9.13.1  |-  A  e. 
_V
Assertion
Ref Expression
tz9.13  |-  E. x  e.  On  A  e.  ( R1 `  x )
Distinct variable group:    x, A

Proof of Theorem tz9.13
Dummy variables  y 
z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tz9.13.1 . . 3  |-  A  e. 
_V
2 setind 7662 . . . 4  |-  ( A. z ( z  C_  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  ->  z  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) } )  ->  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  =  _V )
3 ssel 3334 . . . . . . . 8  |-  ( z 
C_  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  ->  ( w  e.  z  ->  w  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) } ) )
4 vex 2951 . . . . . . . . 9  |-  w  e. 
_V
5 eleq1 2495 . . . . . . . . . 10  |-  ( y  =  w  ->  (
y  e.  ( R1
`  x )  <->  w  e.  ( R1 `  x ) ) )
65rexbidv 2718 . . . . . . . . 9  |-  ( y  =  w  ->  ( E. x  e.  On  y  e.  ( R1 `  x )  <->  E. x  e.  On  w  e.  ( R1 `  x ) ) )
74, 6elab 3074 . . . . . . . 8  |-  ( w  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  <->  E. x  e.  On  w  e.  ( R1 `  x ) )
83, 7syl6ib 218 . . . . . . 7  |-  ( z 
C_  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  ->  ( w  e.  z  ->  E. x  e.  On  w  e.  ( R1 `  x ) ) )
98ralrimiv 2780 . . . . . 6  |-  ( z 
C_  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  ->  A. w  e.  z  E. x  e.  On  w  e.  ( R1 `  x
) )
10 vex 2951 . . . . . . 7  |-  z  e. 
_V
1110tz9.12 7705 . . . . . 6  |-  ( A. w  e.  z  E. x  e.  On  w  e.  ( R1 `  x
)  ->  E. x  e.  On  z  e.  ( R1 `  x ) )
129, 11syl 16 . . . . 5  |-  ( z 
C_  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  ->  E. x  e.  On  z  e.  ( R1 `  x
) )
13 eleq1 2495 . . . . . . 7  |-  ( y  =  z  ->  (
y  e.  ( R1
`  x )  <->  z  e.  ( R1 `  x ) ) )
1413rexbidv 2718 . . . . . 6  |-  ( y  =  z  ->  ( E. x  e.  On  y  e.  ( R1 `  x )  <->  E. x  e.  On  z  e.  ( R1 `  x ) ) )
1510, 14elab 3074 . . . . 5  |-  ( z  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  <->  E. x  e.  On  z  e.  ( R1 `  x ) )
1612, 15sylibr 204 . . . 4  |-  ( z 
C_  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  ->  z  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) } )
172, 16mpg 1557 . . 3  |-  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  =  _V
181, 17eleqtrri 2508 . 2  |-  A  e. 
{ y  |  E. x  e.  On  y  e.  ( R1 `  x
) }
19 eleq1 2495 . . . 4  |-  ( y  =  A  ->  (
y  e.  ( R1
`  x )  <->  A  e.  ( R1 `  x ) ) )
2019rexbidv 2718 . . 3  |-  ( y  =  A  ->  ( E. x  e.  On  y  e.  ( R1 `  x )  <->  E. x  e.  On  A  e.  ( R1 `  x ) ) )
211, 20elab 3074 . 2  |-  ( A  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  <->  E. x  e.  On  A  e.  ( R1 `  x ) )
2218, 21mpbi 200 1  |-  E. x  e.  On  A  e.  ( R1 `  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312   Oncon0 4573   ` cfv 5445   R1cr1 7677
This theorem is referenced by:  tz9.13g  7707  elhf2  26064
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  ax-reg 7549  ax-inf2 7585
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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