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Theorem fr0 4372
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 4358 . 2  |-  ( R  Fr  (/)  <->  A. x ( ( x  C_  (/)  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
2 ss0 3485 . . . . 5  |-  ( x 
C_  (/)  ->  x  =  (/) )
32a1d 22 . . . 4  |-  ( x 
C_  (/)  ->  ( -.  E. y  e.  x  {
z  e.  x  |  z R y }  =  (/)  ->  x  =  (/) ) )
43necon1ad 2513 . . 3  |-  ( x 
C_  (/)  ->  ( x  =/=  (/)  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
54imp 418 . 2  |-  ( ( x  C_  (/)  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) )
61, 5mpgbir 1537 1  |-  R  Fr  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    =/= wne 2446   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   class class class wbr 4023    Fr wfr 4349
This theorem is referenced by:  we0  4388  frsn  4760  frfi  7102  ifr0  27653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-fr 4352
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