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

Definition df-fr 4541
Description: Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 4547 and dffr3 5236. (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 4538 . 2  wff  R  Fr  A
4 vx . . . . . . 7  set  x
54cv 1651 . . . . . 6  class  x
65, 1wss 3320 . . . . 5  wff  x  C_  A
7 c0 3628 . . . . . 6  class  (/)
85, 7wne 2599 . . . . 5  wff  x  =/=  (/)
96, 8wa 359 . . . 4  wff  ( x 
C_  A  /\  x  =/=  (/) )
10 vz . . . . . . . . 9  set  z
1110cv 1651 . . . . . . . 8  class  z
12 vy . . . . . . . . 9  set  y
1312cv 1651 . . . . . . . 8  class  y
1411, 13, 2wbr 4212 . . . . . . 7  wff  z R y
1514wn 3 . . . . . 6  wff  -.  z R y
1615, 10, 5wral 2705 . . . . 5  wff  A. z  e.  x  -.  z R y
1716, 12, 5wrex 2706 . . . 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 1549 . 2  wff  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )
203, 19wb 177 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  4544  dffr2  4547  frss  4549  freq1  4552  nffr  4556  frinxp  4943  frsn  4948  f1oweALT  6074  frxp  6456  frfi  7352  fpwwe2lem12  8516  fpwwe2lem13  8517  dffr5  25376  dfon2lem9  25418  fnwe2  27128  bnj1154  29368
  Copyright terms: Public domain W3C validator