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Theorem tridc 6796
Description: A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.)
Hypotheses
Ref Expression
tridc.po  |-  ( ph  ->  R  Po  A )
tridc.tri  |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )
tridc.b  |-  ( ph  ->  B  e.  A )
tridc.c  |-  ( ph  ->  C  e.  A )
Assertion
Ref Expression
tridc  |-  ( ph  -> DECID  B R C )
Distinct variable groups:    x, A, y   
x, B, y    y, C    x, R, y
Allowed substitution hints:    ph( x, y)    C( x)

Proof of Theorem tridc
StepHypRef Expression
1 simpr 109 . . . 4  |-  ( (
ph  /\  B R C )  ->  B R C )
21orcd 722 . . 3  |-  ( (
ph  /\  B R C )  ->  ( B R C  \/  -.  B R C ) )
3 df-dc 820 . . 3  |-  (DECID  B R C  <->  ( B R C  \/  -.  B R C ) )
42, 3sylibr 133 . 2  |-  ( (
ph  /\  B R C )  -> DECID  B R C )
5 tridc.po . . . . . . 7  |-  ( ph  ->  R  Po  A )
6 tridc.c . . . . . . 7  |-  ( ph  ->  C  e.  A )
7 poirr 4232 . . . . . . 7  |-  ( ( R  Po  A  /\  C  e.  A )  ->  -.  C R C )
85, 6, 7syl2anc 408 . . . . . 6  |-  ( ph  ->  -.  C R C )
98adantr 274 . . . . 5  |-  ( (
ph  /\  B  =  C )  ->  -.  C R C )
10 simpr 109 . . . . . 6  |-  ( (
ph  /\  B  =  C )  ->  B  =  C )
1110breq1d 3942 . . . . 5  |-  ( (
ph  /\  B  =  C )  ->  ( B R C  <->  C R C ) )
129, 11mtbird 662 . . . 4  |-  ( (
ph  /\  B  =  C )  ->  -.  B R C )
1312olcd 723 . . 3  |-  ( (
ph  /\  B  =  C )  ->  ( B R C  \/  -.  B R C ) )
1413, 3sylibr 133 . 2  |-  ( (
ph  /\  B  =  C )  -> DECID  B R C )
15 tridc.b . . . . . . 7  |-  ( ph  ->  B  e.  A )
16 po2nr 4234 . . . . . . 7  |-  ( ( R  Po  A  /\  ( C  e.  A  /\  B  e.  A
) )  ->  -.  ( C R B  /\  B R C ) )
175, 6, 15, 16syl12anc 1214 . . . . . 6  |-  ( ph  ->  -.  ( C R B  /\  B R C ) )
1817adantr 274 . . . . 5  |-  ( (
ph  /\  C R B )  ->  -.  ( C R B  /\  B R C ) )
19 simplr 519 . . . . . 6  |-  ( ( ( ph  /\  C R B )  /\  B R C )  ->  C R B )
20 simpr 109 . . . . . 6  |-  ( ( ( ph  /\  C R B )  /\  B R C )  ->  B R C )
2119, 20jca 304 . . . . 5  |-  ( ( ( ph  /\  C R B )  /\  B R C )  ->  ( C R B  /\  B R C ) )
2218, 21mtand 654 . . . 4  |-  ( (
ph  /\  C R B )  ->  -.  B R C )
2322olcd 723 . . 3  |-  ( (
ph  /\  C R B )  ->  ( B R C  \/  -.  B R C ) )
2423, 3sylibr 133 . 2  |-  ( (
ph  /\  C R B )  -> DECID  B R C )
25 tridc.tri . . 3  |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )
26 breq1 3935 . . . . 5  |-  ( x  =  B  ->  (
x R y  <->  B R
y ) )
27 eqeq1 2146 . . . . 5  |-  ( x  =  B  ->  (
x  =  y  <->  B  =  y ) )
28 breq2 3936 . . . . 5  |-  ( x  =  B  ->  (
y R x  <->  y R B ) )
2926, 27, 283orbi123d 1289 . . . 4  |-  ( x  =  B  ->  (
( x R y  \/  x  =  y  \/  y R x )  <->  ( B R y  \/  B  =  y  \/  y R B ) ) )
30 breq2 3936 . . . . 5  |-  ( y  =  C  ->  ( B R y  <->  B R C ) )
31 eqeq2 2149 . . . . 5  |-  ( y  =  C  ->  ( B  =  y  <->  B  =  C ) )
32 breq1 3935 . . . . 5  |-  ( y  =  C  ->  (
y R B  <->  C R B ) )
3330, 31, 323orbi123d 1289 . . . 4  |-  ( y  =  C  ->  (
( B R y  \/  B  =  y  \/  y R B )  <->  ( B R C  \/  B  =  C  \/  C R B ) ) )
3429, 33rspc2va 2803 . . 3  |-  ( ( ( B  e.  A  /\  C  e.  A
)  /\  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
3515, 6, 25, 34syl21anc 1215 . 2  |-  ( ph  ->  ( B R C  \/  B  =  C  \/  C R B ) )
364, 14, 24, 35mpjao3dan 1285 1  |-  ( ph  -> DECID  B R C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 697  DECID wdc 819    \/ w3o 961    = wceq 1331    e. wcel 1480   A.wral 2416   class class class wbr 3932    Po wpo 4219
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 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3933  df-po 4221
This theorem is referenced by:  fimax2gtrilemstep  6797
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