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Theorem tz9.13 7458
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
Dummy variables  y 
z  w are mutually distinct and distinct from all other variables.

Proof of Theorem tz9.13
StepHypRef Expression
1 tz9.13.1 . . 3  |-  A  e. 
_V
2 setind 7414 . . . 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 3175 . . . . . . . 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 2792 . . . . . . . . 9  |-  w  e. 
_V
5 eleq1 2344 . . . . . . . . . 10  |-  ( y  =  w  ->  (
y  e.  ( R1
`  x )  <->  w  e.  ( R1 `  x ) ) )
65rexbidv 2565 . . . . . . . . 9  |-  ( y  =  w  ->  ( E. x  e.  On  y  e.  ( R1 `  x )  <->  E. x  e.  On  w  e.  ( R1 `  x ) ) )
74, 6elab 2915 . . . . . . . 8  |-  ( w  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  <->  E. x  e.  On  w  e.  ( R1 `  x ) )
83, 7syl6ib 219 . . . . . . 7  |-  ( z 
C_  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  ->  ( w  e.  z  ->  E. x  e.  On  w  e.  ( R1 `  x ) ) )
98ralrimiv 2626 . . . . . 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 2792 . . . . . . 7  |-  z  e. 
_V
1110tz9.12 7457 . . . . . 6  |-  ( A. w  e.  z  E. x  e.  On  w  e.  ( R1 `  x
)  ->  E. x  e.  On  z  e.  ( R1 `  x ) )
129, 11syl 17 . . . . 5  |-  ( z 
C_  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  ->  E. x  e.  On  z  e.  ( R1 `  x
) )
13 eleq1 2344 . . . . . . 7  |-  ( y  =  z  ->  (
y  e.  ( R1
`  x )  <->  z  e.  ( R1 `  x ) ) )
1413rexbidv 2565 . . . . . 6  |-  ( y  =  z  ->  ( E. x  e.  On  y  e.  ( R1 `  x )  <->  E. x  e.  On  z  e.  ( R1 `  x ) ) )
1510, 14elab 2915 . . . . 5  |-  ( z  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  <->  E. x  e.  On  z  e.  ( R1 `  x ) )
1612, 15sylibr 205 . . . 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 1536 . . 3  |-  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  =  _V
181, 17eleqtrri 2357 . 2  |-  A  e. 
{ y  |  E. x  e.  On  y  e.  ( R1 `  x
) }
19 eleq1 2344 . . . 4  |-  ( y  =  A  ->  (
y  e.  ( R1
`  x )  <->  A  e.  ( R1 `  x ) ) )
2019rexbidv 2565 . . 3  |-  ( y  =  A  ->  ( E. x  e.  On  y  e.  ( R1 `  x )  <->  E. x  e.  On  A  e.  ( R1 `  x ) ) )
211, 20elab 2915 . 2  |-  ( A  e.  { y  |  E. x  e.  On  y  e.  ( R1 `  x ) }  <->  E. x  e.  On  A  e.  ( R1 `  x ) )
2218, 21mpbi 201 1  |-  E. x  e.  On  A  e.  ( R1 `  x )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   {cab 2270   A.wral 2544   E.wrex 2545   _Vcvv 2789    C_ wss 3153   Oncon0 4391   ` cfv 5221   R1cr1 7429
This theorem is referenced by:  tz9.13g  7459
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  ax-reg 7301  ax-inf2 7337
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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