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Theorem dfepfr 4378
Description: An alternate way of saying that the epsilon relation is well-founded. (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)
Assertion
Ref Expression
dfepfr  |-  (  _E  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  y
)  =  (/) ) )
Distinct variable group:    x, y, A

Proof of Theorem dfepfr
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dffr2 4358 . 2  |-  (  _E  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z  _E  y }  =  (/) ) )
2 epel 4308 . . . . . . . . 9  |-  ( z  _E  y  <->  z  e.  y )
32a1i 10 . . . . . . . 8  |-  ( z  e.  x  ->  (
z  _E  y  <->  z  e.  y ) )
43rabbiia 2778 . . . . . . 7  |-  { z  e.  x  |  z  _E  y }  =  { z  e.  x  |  z  e.  y }
5 dfin5 3160 . . . . . . 7  |-  ( x  i^i  y )  =  { z  e.  x  |  z  e.  y }
64, 5eqtr4i 2306 . . . . . 6  |-  { z  e.  x  |  z  _E  y }  =  ( x  i^i  y
)
76eqeq1i 2290 . . . . 5  |-  ( { z  e.  x  |  z  _E  y }  =  (/)  <->  ( x  i^i  y )  =  (/) )
87rexbii 2568 . . . 4  |-  ( E. y  e.  x  {
z  e.  x  |  z  _E  y }  =  (/)  <->  E. y  e.  x  ( x  i^i  y
)  =  (/) )
98imbi2i 303 . . 3  |-  ( ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z  _E  y }  =  (/) )  <->  ( (
x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  y )  =  (/) ) )
109albii 1553 . 2  |-  ( A. x ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z  _E  y }  =  (/) ) 
<-> 
A. x ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  y )  =  (/) ) )
111, 10bitri 240 1  |-  (  _E  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  y
)  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   (/)c0 3455   class class class wbr 4023    _E cep 4303    Fr wfr 4349
This theorem is referenced by:  onfr  4431  zfregfr  7316  onfrALTlem3  28309  onfrALT  28314  onfrALTlem3VD  28663  onfrALTVD  28667
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-fr 4352
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