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Theorem trirec0xor 13386
Description: Version of trirec0 13385 with exclusive-or.

The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.)

Assertion
Ref Expression
trirec0xor  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  <->  A. x  e.  RR  ( E. z  e.  RR  ( x  x.  z )  =  1 
\/_  x  =  0 ) )
Distinct variable group:    x, y, z

Proof of Theorem trirec0xor
StepHypRef Expression
1 trirec0 13385 . 2  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  <->  A. x  e.  RR  ( E. z  e.  RR  ( x  x.  z )  =  1  \/  x  =  0 ) )
2 1ne0 8807 . . . . . . . 8  |-  1  =/=  0
32nesymi 2354 . . . . . . 7  |-  -.  0  =  1
4 simpr 109 . . . . . . . . . . 11  |-  ( ( ( x  x.  z
)  =  1  /\  x  =  0 )  ->  x  =  0 )
54oveq1d 5792 . . . . . . . . . 10  |-  ( ( ( x  x.  z
)  =  1  /\  x  =  0 )  ->  ( x  x.  z )  =  ( 0  x.  z ) )
6 mul02lem2 8169 . . . . . . . . . 10  |-  ( z  e.  RR  ->  (
0  x.  z )  =  0 )
75, 6sylan9eqr 2194 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( ( x  x.  z )  =  1  /\  x  =  0 ) )  ->  (
x  x.  z )  =  0 )
8 simprl 520 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( ( x  x.  z )  =  1  /\  x  =  0 ) )  ->  (
x  x.  z )  =  1 )
97, 8eqtr3d 2174 . . . . . . . 8  |-  ( ( z  e.  RR  /\  ( ( x  x.  z )  =  1  /\  x  =  0 ) )  ->  0  =  1 )
109rexlimiva 2544 . . . . . . 7  |-  ( E. z  e.  RR  (
( x  x.  z
)  =  1  /\  x  =  0 )  ->  0  =  1 )
113, 10mto 651 . . . . . 6  |-  -.  E. z  e.  RR  (
( x  x.  z
)  =  1  /\  x  =  0 )
12 r19.41v 2587 . . . . . 6  |-  ( E. z  e.  RR  (
( x  x.  z
)  =  1  /\  x  =  0 )  <-> 
( E. z  e.  RR  ( x  x.  z )  =  1  /\  x  =  0 ) )
1311, 12mtbi 659 . . . . 5  |-  -.  ( E. z  e.  RR  ( x  x.  z
)  =  1  /\  x  =  0 )
1413biantru 300 . . . 4  |-  ( ( E. z  e.  RR  ( x  x.  z
)  =  1  \/  x  =  0 )  <-> 
( ( E. z  e.  RR  ( x  x.  z )  =  1  \/  x  =  0 )  /\  -.  ( E. z  e.  RR  ( x  x.  z
)  =  1  /\  x  =  0 ) ) )
15 df-xor 1354 . . . 4  |-  ( ( E. z  e.  RR  ( x  x.  z
)  =  1  \/_  x  =  0 )  <-> 
( ( E. z  e.  RR  ( x  x.  z )  =  1  \/  x  =  0 )  /\  -.  ( E. z  e.  RR  ( x  x.  z
)  =  1  /\  x  =  0 ) ) )
1614, 15bitr4i 186 . . 3  |-  ( ( E. z  e.  RR  ( x  x.  z
)  =  1  \/  x  =  0 )  <-> 
( E. z  e.  RR  ( x  x.  z )  =  1 
\/_  x  =  0 ) )
1716ralbii 2441 . 2  |-  ( A. x  e.  RR  ( E. z  e.  RR  ( x  x.  z
)  =  1  \/  x  =  0 )  <->  A. x  e.  RR  ( E. z  e.  RR  ( x  x.  z
)  =  1  \/_  x  =  0 ) )
181, 17bitri 183 1  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  <->  A. x  e.  RR  ( E. z  e.  RR  ( x  x.  z )  =  1 
\/_  x  =  0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    \/ wo 697    \/ w3o 961    = wceq 1331    \/_ wxo 1353    e. wcel 1480   A.wral 2416   E.wrex 2417   class class class wbr 3932  (class class class)co 5777   RRcr 7638   0cc0 7639   1c1 7640    x. cmul 7644    < clt 7819
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-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-cnex 7730  ax-resscn 7731  ax-1cn 7732  ax-1re 7733  ax-icn 7734  ax-addcl 7735  ax-addrcl 7736  ax-mulcl 7737  ax-mulrcl 7738  ax-addcom 7739  ax-mulcom 7740  ax-addass 7741  ax-mulass 7742  ax-distr 7743  ax-i2m1 7744  ax-0lt1 7745  ax-1rid 7746  ax-0id 7747  ax-rnegex 7748  ax-precex 7749  ax-cnre 7750  ax-pre-ltirr 7751  ax-pre-ltwlin 7752  ax-pre-lttrn 7753  ax-pre-apti 7754  ax-pre-ltadd 7755  ax-pre-mulgt0 7756  ax-pre-mulext 7757
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-xor 1354  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-id 4218  df-po 4221  df-iso 4222  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-iota 5091  df-fun 5128  df-fv 5134  df-riota 5733  df-ov 5780  df-oprab 5781  df-mpo 5782  df-pnf 7821  df-mnf 7822  df-xr 7823  df-ltxr 7824  df-le 7825  df-sub 7954  df-neg 7955  df-reap 8356  df-ap 8363  df-div 8452
This theorem is referenced by: (None)
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