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Definition df-fr 4827
Description: Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 4833 and dffr3 5357. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
df-fr  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
Distinct variable groups:    x, y,
z, R    x, A, y, z

Detailed syntax breakdown of Definition df-fr
StepHypRef Expression
1 cA . . 3  class  A
2 cR . . 3  class  R
31, 2wfr 4824 . 2  wff  R  Fr  A
4 vx . . . . . . 7  setvar  x
54cv 1397 . . . . . 6  class  x
65, 1wss 3461 . . . . 5  wff  x  C_  A
7 c0 3783 . . . . . 6  class  (/)
85, 7wne 2649 . . . . 5  wff  x  =/=  (/)
96, 8wa 367 . . . 4  wff  ( x 
C_  A  /\  x  =/=  (/) )
10 vz . . . . . . . . 9  setvar  z
1110cv 1397 . . . . . . . 8  class  z
12 vy . . . . . . . . 9  setvar  y
1312cv 1397 . . . . . . . 8  class  y
1411, 13, 2wbr 4439 . . . . . . 7  wff  z R y
1514wn 3 . . . . . 6  wff  -.  z R y
1615, 10, 5wral 2804 . . . . 5  wff  A. z  e.  x  -.  z R y
1716, 12, 5wrex 2805 . . . 4  wff  E. y  e.  x  A. z  e.  x  -.  z R y
189, 17wi 4 . . 3  wff  ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )
1918, 4wal 1396 . 2  wff  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )
203, 19wb 184 1  wff  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
Colors of variables: wff setvar class
This definition is referenced by:  fri  4830  dffr2  4833  frss  4835  freq1  4838  nffr  4842  frinxp  5054  frsn  5059  f1oweALT  6757  frxp  6883  frfi  7757  fpwwe2lem12  9008  fpwwe2lem13  9009  dffr5  29423  dfon2lem9  29463  fin2so  30280  fnwe2  31238  bnj1154  34456
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