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Theorem dfepfr 4394
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 4374 . 2  |-  (  _E  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z  _E  y }  =  (/) ) )
2 epel 4324 . . . . . . . . 9  |-  ( z  _E  y  <->  z  e.  y )
32a1i 10 . . . . . . . 8  |-  ( z  e.  x  ->  (
z  _E  y  <->  z  e.  y ) )
43rabbiia 2791 . . . . . . 7  |-  { z  e.  x  |  z  _E  y }  =  { z  e.  x  |  z  e.  y }
5 dfin5 3173 . . . . . . 7  |-  ( x  i^i  y )  =  { z  e.  x  |  z  e.  y }
64, 5eqtr4i 2319 . . . . . 6  |-  { z  e.  x  |  z  _E  y }  =  ( x  i^i  y
)
76eqeq1i 2303 . . . . 5  |-  ( { z  e.  x  |  z  _E  y }  =  (/)  <->  ( x  i^i  y )  =  (/) )
87rexbii 2581 . . . 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 1556 . 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 1530    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   (/)c0 3468   class class class wbr 4039    _E cep 4319    Fr wfr 4365
This theorem is referenced by:  onfr  4447  zfregfr  7332  onfrALTlem3  28608  onfrALT  28613  onfrALTlem3VD  28979  onfrALTVD  28983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-eprel 4321  df-fr 4368
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