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Theorem fr0 4847
Description: Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
Assertion
Ref Expression
fr0  |-  R  Fr  (/)

Proof of Theorem fr0
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffr2 4833 . 2  |-  ( R  Fr  (/)  <->  A. x ( ( x  C_  (/)  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
2 ss0 3815 . . . . 5  |-  ( x 
C_  (/)  ->  x  =  (/) )
32a1d 25 . . . 4  |-  ( x 
C_  (/)  ->  ( -.  E. y  e.  x  {
z  e.  x  |  z R y }  =  (/)  ->  x  =  (/) ) )
43necon1ad 2670 . . 3  |-  ( x 
C_  (/)  ->  ( x  =/=  (/)  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
54imp 427 . 2  |-  ( ( x  C_  (/)  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) )
61, 5mpgbir 1627 1  |-  R  Fr  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    =/= wne 2649   E.wrex 2805   {crab 2808    C_ wss 3461   (/)c0 3783   class class class wbr 4439    Fr wfr 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-fr 4827
This theorem is referenced by:  we0  4863  frsn  5059  frfi  7757  ifr0  31603
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