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Theorem fr0 4561
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 4547 . 2  |-  ( R  Fr  (/)  <->  A. x ( ( x  C_  (/)  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
2 ss0 3658 . . . . 5  |-  ( x 
C_  (/)  ->  x  =  (/) )
32a1d 23 . . . 4  |-  ( x 
C_  (/)  ->  ( -.  E. y  e.  x  {
z  e.  x  |  z R y }  =  (/)  ->  x  =  (/) ) )
43necon1ad 2671 . . 3  |-  ( x 
C_  (/)  ->  ( x  =/=  (/)  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
54imp 419 . 2  |-  ( ( x  C_  (/)  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) )
61, 5mpgbir 1559 1  |-  R  Fr  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    =/= wne 2599   E.wrex 2706   {crab 2709    C_ wss 3320   (/)c0 3628   class class class wbr 4212    Fr wfr 4538
This theorem is referenced by:  we0  4577  frsn  4948  frfi  7352  ifr0  27629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-fr 4541
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