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Theorem dfepfr 4853
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 4833 . 2  |-  (  _E  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z  _E  y }  =  (/) ) )
2 epel 4783 . . . . . . . . 9  |-  ( z  _E  y  <->  z  e.  y )
32a1i 11 . . . . . . . 8  |-  ( z  e.  x  ->  (
z  _E  y  <->  z  e.  y ) )
43rabbiia 3095 . . . . . . 7  |-  { z  e.  x  |  z  _E  y }  =  { z  e.  x  |  z  e.  y }
5 dfin5 3469 . . . . . . 7  |-  ( x  i^i  y )  =  { z  e.  x  |  z  e.  y }
64, 5eqtr4i 2486 . . . . . 6  |-  { z  e.  x  |  z  _E  y }  =  ( x  i^i  y
)
76eqeq1i 2461 . . . . 5  |-  ( { z  e.  x  |  z  _E  y }  =  (/)  <->  ( x  i^i  y )  =  (/) )
87rexbii 2956 . . . 4  |-  ( E. y  e.  x  {
z  e.  x  |  z  _E  y }  =  (/)  <->  E. y  e.  x  ( x  i^i  y
)  =  (/) )
98imbi2i 310 . . 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 1645 . 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 249 1  |-  (  _E  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  y
)  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398    =/= wne 2649   E.wrex 2805   {crab 2808    i^i cin 3460    C_ wss 3461   (/)c0 3783   class class class wbr 4439    _E cep 4778    Fr wfr 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-eprel 4780  df-fr 4827
This theorem is referenced by:  onfr  4906  zfregfr  8020  onfrALTlem3  33710  onfrALT  33715  onfrALTlem3VD  34088  onfrALTVD  34092
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