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Theorem fr0 4783
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 4769 . 2  |-  ( R  Fr  (/)  <->  A. x ( ( x  C_  (/)  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
2 ss0 3752 . . . . 5  |-  ( x 
C_  (/)  ->  x  =  (/) )
32a1d 25 . . . 4  |-  ( x 
C_  (/)  ->  ( -.  E. y  e.  x  {
z  e.  x  |  z R y }  =  (/)  ->  x  =  (/) ) )
43necon1ad 2661 . . 3  |-  ( x 
C_  (/)  ->  ( x  =/=  (/)  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
54imp 429 . 2  |-  ( ( x  C_  (/)  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) )
61, 5mpgbir 1596 1  |-  R  Fr  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    =/= wne 2641   E.wrex 2793   {crab 2796    C_ wss 3412   (/)c0 3721   class class class wbr 4376    Fr wfr 4760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-dif 3415  df-in 3419  df-ss 3426  df-nul 3722  df-fr 4763
This theorem is referenced by:  we0  4799  frsn  4993  frfi  7644  ifr0  29830
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