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Theorem onntri24 7219
Description: Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
Assertion
Ref Expression
onntri24  |-  ( -. 
-.  A. x  e.  On  A. y  e.  On  (
x  C_  y  \/  y  C_  x )  ->  A. x  e.  On  A. y  e.  On  -.  -.  ( x  C_  y  \/  y  C_  x ) )

Proof of Theorem onntri24
StepHypRef Expression
1 nnral 2460 . 2  |-  ( -. 
-.  A. x  e.  On  A. y  e.  On  (
x  C_  y  \/  y  C_  x )  ->  A. x  e.  On  -.  -.  A. y  e.  On  ( x  C_  y  \/  y  C_  x ) )
2 nnral 2460 . . 3  |-  ( -. 
-.  A. y  e.  On  ( x  C_  y  \/  y  C_  x )  ->  A. y  e.  On  -.  -.  ( x  C_  y  \/  y  C_  x ) )
32ralimi 2533 . 2  |-  ( A. x  e.  On  -.  -.  A. y  e.  On  ( x  C_  y  \/  y  C_  x )  ->  A. x  e.  On  A. y  e.  On  -.  -.  ( x  C_  y  \/  y  C_  x ) )
41, 3syl 14 1  |-  ( -. 
-.  A. x  e.  On  A. y  e.  On  (
x  C_  y  \/  y  C_  x )  ->  A. x  e.  On  A. y  e.  On  -.  -.  ( x  C_  y  \/  y  C_  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 703   A.wral 2448    C_ wss 3121   Oncon0 4348
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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-ral 2453  df-rex 2454
This theorem is referenced by:  onntri2or  7223
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