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Theorem dfepfr 4568
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 4548 . 2  |-  (  _E  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z  _E  y }  =  (/) ) )
2 epel 4498 . . . . . . . . 9  |-  ( z  _E  y  <->  z  e.  y )
32a1i 11 . . . . . . . 8  |-  ( z  e.  x  ->  (
z  _E  y  <->  z  e.  y ) )
43rabbiia 2947 . . . . . . 7  |-  { z  e.  x  |  z  _E  y }  =  { z  e.  x  |  z  e.  y }
5 dfin5 3329 . . . . . . 7  |-  ( x  i^i  y )  =  { z  e.  x  |  z  e.  y }
64, 5eqtr4i 2460 . . . . . 6  |-  { z  e.  x  |  z  _E  y }  =  ( x  i^i  y
)
76eqeq1i 2444 . . . . 5  |-  ( { z  e.  x  |  z  _E  y }  =  (/)  <->  ( x  i^i  y )  =  (/) )
87rexbii 2731 . . . 4  |-  ( E. y  e.  x  {
z  e.  x  |  z  _E  y }  =  (/)  <->  E. y  e.  x  ( x  i^i  y
)  =  (/) )
98imbi2i 305 . . 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 1576 . 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 242 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 178    /\ wa 360   A.wal 1550    = wceq 1653    =/= wne 2600   E.wrex 2707   {crab 2710    i^i cin 3320    C_ wss 3321   (/)c0 3629   class class class wbr 4213    _E cep 4493    Fr wfr 4539
This theorem is referenced by:  onfr  4621  zfregfr  7571  onfrALTlem3  28631  onfrALT  28636  onfrALTlem3VD  29000  onfrALTVD  29004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-eprel 4495  df-fr 4542
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