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Theorem dffr2 4539
Description: Alternate definition of well-founded relation. Similar to Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Mario Carneiro, 23-Jun-2015.)
Assertion
Ref Expression
dffr2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
Distinct variable groups:    x, y,
z, A    x, R, y, z

Proof of Theorem dffr2
StepHypRef Expression
1 df-fr 4533 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
2 rabeq0 3641 . . . . 5  |-  ( { z  e.  x  |  z R y }  =  (/)  <->  A. z  e.  x  -.  z R y )
32rexbii 2722 . . . 4  |-  ( E. y  e.  x  {
z  e.  x  |  z R y }  =  (/)  <->  E. y  e.  x  A. z  e.  x  -.  z R y )
43imbi2i 304 . . 3  |-  ( ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) )  <->  ( (
x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
54albii 1575 . 2  |-  ( A. x ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) )  <->  A. x ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
61, 5bitr4i 244 1  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701    C_ wss 3312   (/)c0 3620   class class class wbr 4204    Fr wfr 4530
This theorem is referenced by:  fr0  4553  dfepfr  4559  dffr3  5227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-nul 3621  df-fr 4533
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