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Theorem tridceq 14460
Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 14447 and redcwlpo 14459). Thus, this is an analytic analogue to lpowlpo 7160. (Contributed by Jim Kingdon, 24-Jul-2024.)
Assertion
Ref Expression
tridceq  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  ->  A. x  e.  RR  A. y  e.  RR DECID  x  =  y )
Distinct variable group:    x, y

Proof of Theorem tridceq
StepHypRef Expression
1 ltne 8032 . . . . . . 7  |-  ( ( x  e.  RR  /\  x  <  y )  -> 
y  =/=  x )
21ex 115 . . . . . 6  |-  ( x  e.  RR  ->  (
x  <  y  ->  y  =/=  x ) )
32adantr 276 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  ->  y  =/=  x ) )
4 olc 711 . . . . . 6  |-  ( x  =/=  y  ->  (
x  =  y  \/  x  =/=  y ) )
5 necom 2431 . . . . . 6  |-  ( y  =/=  x  <->  x  =/=  y )
6 dcne 2358 . . . . . 6  |-  (DECID  x  =  y  <->  ( x  =  y  \/  x  =/=  y ) )
74, 5, 63imtr4i 201 . . . . 5  |-  ( y  =/=  x  -> DECID  x  =  y
)
83, 7syl6 33 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  -> DECID  x  =  y ) )
9 orc 712 . . . . . 6  |-  ( x  =  y  ->  (
x  =  y  \/  x  =/=  y ) )
109, 6sylibr 134 . . . . 5  |-  ( x  =  y  -> DECID  x  =  y
)
1110a1i 9 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  y  -> DECID 
x  =  y ) )
12 ltne 8032 . . . . . . 7  |-  ( ( y  e.  RR  /\  y  <  x )  ->  x  =/=  y )
1312ex 115 . . . . . 6  |-  ( y  e.  RR  ->  (
y  <  x  ->  x  =/=  y ) )
1413adantl 277 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( y  <  x  ->  x  =/=  y ) )
154, 6sylibr 134 . . . . 5  |-  ( x  =/=  y  -> DECID  x  =  y
)
1614, 15syl6 33 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( y  <  x  -> DECID  x  =  y ) )
178, 11, 163jaod 1304 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  < 
y  \/  x  =  y  \/  y  < 
x )  -> DECID  x  =  y
) )
1817ralimdva 2544 . 2  |-  ( x  e.  RR  ->  ( A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x )  ->  A. y  e.  RR DECID  x  =  y ) )
1918ralimia 2538 1  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  ->  A. x  e.  RR  A. y  e.  RR DECID  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 708  DECID wdc 834    \/ w3o 977    e. wcel 2148    =/= wne 2347   A.wral 2455   class class class wbr 4000   RRcr 7801    < clt 7982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-pre-ltirr 7914
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4629  df-pnf 7984  df-mnf 7985  df-ltxr 7987
This theorem is referenced by:  dcapnconstALT  14465
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