MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-fr Structured version   Unicode version

Definition df-fr 4542
Description: Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 4548 and dffr3 5237. (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 4539 . 2  wff  R  Fr  A
4 vx . . . . . . 7  set  x
54cv 1652 . . . . . 6  class  x
65, 1wss 3321 . . . . 5  wff  x  C_  A
7 c0 3629 . . . . . 6  class  (/)
85, 7wne 2600 . . . . 5  wff  x  =/=  (/)
96, 8wa 360 . . . 4  wff  ( x 
C_  A  /\  x  =/=  (/) )
10 vz . . . . . . . . 9  set  z
1110cv 1652 . . . . . . . 8  class  z
12 vy . . . . . . . . 9  set  y
1312cv 1652 . . . . . . . 8  class  y
1411, 13, 2wbr 4213 . . . . . . 7  wff  z R y
1514wn 3 . . . . . 6  wff  -.  z R y
1615, 10, 5wral 2706 . . . . 5  wff  A. z  e.  x  -.  z R y
1716, 12, 5wrex 2707 . . . 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 1550 . 2  wff  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )
203, 19wb 178 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 set class
This definition is referenced by:  fri  4545  dffr2  4548  frss  4550  freq1  4553  nffr  4557  frinxp  4944  frsn  4949  f1oweALT  6075  frxp  6457  frfi  7353  fpwwe2lem12  8517  fpwwe2lem13  8518  dffr5  25377  dfon2lem9  25419  fnwe2  27129  bnj1154  29369
  Copyright terms: Public domain W3C validator