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Theorem exmidontri 7216
Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
Assertion
Ref Expression
exmidontri  |-  (EXMID  <->  A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
Distinct variable group:    x, y

Proof of Theorem exmidontri
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 exmidontriim 7202 . 2  |-  (EXMID  ->  A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
2 ontriexmidim 4506 . . . 4  |-  ( A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x )  -> DECID  z  =  { (/)
} )
32adantr 274 . . 3  |-  ( ( A. x  e.  On  A. y  e.  On  (
x  e.  y  \/  x  =  y  \/  y  e.  x )  /\  z  C_  { (/) } )  -> DECID  z  =  { (/)
} )
43exmid1dc 4186 . 2  |-  ( A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x )  -> EXMID )
51, 4impbii 125 1  |-  (EXMID  <->  A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104  DECID wdc 829    \/ w3o 972    = wceq 1348   A.wral 2448    C_ wss 3121   (/)c0 3414   {csn 3583  EXMIDwem 4180   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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-uni 3797  df-tr 4088  df-exmid 4181  df-iord 4351  df-on 4353  df-suc 4356
This theorem is referenced by: (None)
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