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Theorem fr0 4388
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 4374 . 2  |-  ( R  Fr  (/)  <->  A. x ( ( x  C_  (/)  /\  x  =/=  (/) )  ->  E. y  e.  x  { z  e.  x  |  z R y }  =  (/) ) )
2 ss0 3498 . . . . 5  |-  ( x 
C_  (/)  ->  x  =  (/) )
32a1d 22 . . . 4  |-  ( x 
C_  (/)  ->  ( -.  E. y  e.  x  {
z  e.  x  |  z R y }  =  (/)  ->  x  =  (/) ) )
43necon1ad 2526 . . 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 1540 1  |-  R  Fr  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    =/= wne 2459   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   class class class wbr 4039    Fr wfr 4365
This theorem is referenced by:  we0  4404  frsn  4776  frfi  7118  ifr0  27756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-fr 4368
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