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Theorem dfepfr 4814
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 4794 . 2  |-  (  _E  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z  _E  y }  =  (/) ) )
2 epel 4744 . . . . . . . . 9  |-  ( z  _E  y  <->  z  e.  y )
32a1i 11 . . . . . . . 8  |-  ( z  e.  x  ->  (
z  _E  y  <->  z  e.  y ) )
43rabbiia 3067 . . . . . . 7  |-  { z  e.  x  |  z  _E  y }  =  { z  e.  x  |  z  e.  y }
5 dfin5 3445 . . . . . . 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 2862 . . . 4  |-  ( E. y  e.  x  {
z  e.  x  |  z  _E  y }  =  (/)  <->  E. y  e.  x  ( x  i^i  y
)  =  (/) )
98imbi2i 312 . . 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 1611 . 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 369   A.wal 1368    = wceq 1370    =/= wne 2648   E.wrex 2800   {crab 2803    i^i cin 3436    C_ wss 3437   (/)c0 3746   class class class wbr 4401    _E cep 4739    Fr wfr 4785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-eprel 4741  df-fr 4788
This theorem is referenced by:  onfr  4867  zfregfr  7930  onfrALTlem3  31585  onfrALT  31590  onfrALTlem3VD  31956  onfrALTVD  31960
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