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Theorem tz9.13 7463
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 7419 . . . 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 3174 . . . . . . . 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 2791 . . . . . . . . 9  |-  w  e. 
_V
5 eleq1 2343 . . . . . . . . . 10  |-  ( y  =  w  ->  (
y  e.  ( R1
`  x )  <->  w  e.  ( R1 `  x ) ) )
65rexbidv 2564 . . . . . . . . 9  |-  ( y  =  w  ->  ( E. x  e.  On  y  e.  ( R1 `  x )  <->  E. x  e.  On  w  e.  ( R1 `  x ) ) )
74, 6elab 2914 . . . . . . . 8  |-  ( w  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  <->  E. x  e.  On  w  e.  ( R1 `  x ) )
83, 7syl6ib 217 . . . . . . 7  |-  ( z 
C_  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  ->  ( w  e.  z  ->  E. x  e.  On  w  e.  ( R1 `  x ) ) )
98ralrimiv 2625 . . . . . 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 2791 . . . . . . 7  |-  z  e. 
_V
1110tz9.12 7462 . . . . . 6  |-  ( A. w  e.  z  E. x  e.  On  w  e.  ( R1 `  x
)  ->  E. x  e.  On  z  e.  ( R1 `  x ) )
129, 11syl 15 . . . . 5  |-  ( z 
C_  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  ->  E. x  e.  On  z  e.  ( R1 `  x
) )
13 eleq1 2343 . . . . . . 7  |-  ( y  =  z  ->  (
y  e.  ( R1
`  x )  <->  z  e.  ( R1 `  x ) ) )
1413rexbidv 2564 . . . . . 6  |-  ( y  =  z  ->  ( E. x  e.  On  y  e.  ( R1 `  x )  <->  E. x  e.  On  z  e.  ( R1 `  x ) ) )
1510, 14elab 2914 . . . . 5  |-  ( z  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  <->  E. x  e.  On  z  e.  ( R1 `  x ) )
1612, 15sylibr 203 . . . 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 1535 . . 3  |-  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  =  _V
181, 17eleqtrri 2356 . 2  |-  A  e. 
{ y  |  E. x  e.  On  y  e.  ( R1 `  x
) }
19 eleq1 2343 . . . 4  |-  ( y  =  A  ->  (
y  e.  ( R1
`  x )  <->  A  e.  ( R1 `  x ) ) )
2019rexbidv 2564 . . 3  |-  ( y  =  A  ->  ( E. x  e.  On  y  e.  ( R1 `  x )  <->  E. x  e.  On  A  e.  ( R1 `  x ) ) )
211, 20elab 2914 . 2  |-  ( A  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  <->  E. x  e.  On  A  e.  ( R1 `  x ) )
2218, 21mpbi 199 1  |-  E. x  e.  On  A  e.  ( R1 `  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   Oncon0 4392   ` cfv 5255   R1cr1 7434
This theorem is referenced by:  tz9.13g  7464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436
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