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Theorem we0 4278
Description: Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
Assertion
Ref Expression
we0  |-  R  We  (/)

Proof of Theorem we0
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fr0 4268 . 2  |-  R  Fr  (/)
2 ral0 3459 . 2  |-  A. x  e.  (/)  A. y  e.  (/)  A. z  e.  (/)  ( ( x R y  /\  y R z )  ->  x R z )
3 df-wetr 4251 . 2  |-  ( R  We  (/)  <->  ( R  Fr  (/) 
/\  A. x  e.  (/)  A. y  e.  (/)  A. z  e.  (/)  ( ( x R y  /\  y R z )  ->  x R z ) ) )
41, 2, 3mpbir2an 926 1  |-  R  We  (/)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wral 2414   (/)c0 3358   class class class wbr 3924    Fr wfr 4245    We wwe 4247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359  df-frfor 4248  df-frind 4249  df-wetr 4251
This theorem is referenced by: (None)
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