Theorem tridceq 13945

Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 13932 and redcwlpo 13944). Thus, this is an analytic analogue to lpowlpo 7132. (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

Step	Hyp	Ref	Expression
1		ltne 7983	. . . . . . 7 |- ( ( x e. RR /\ x < y ) -> y =/= x )
2	1	ex 114	. . . . . 6 |- ( x e. RR -> ( x < y -> y =/= x ) )
3	2	adantr 274	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( x < y -> y =/= x ) )
4		olc 701	. . . . . 6 |- ( x =/= y -> ( x = y \/ x =/= y ) )
5		necom 2420	. . . . . 6 |- ( y =/= x <-> x =/= y )
6		dcne 2347	. . . . . 6 |- (DECID x = y <-> ( x = y \/ x =/= y ) )
7	4, 5, 6	3imtr4i 200	. . . . 5 |- ( y =/= x -> DECID x = y )
8	3, 7	syl6 33	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x < y -> DECID x = y ) )
9		orc 702	. . . . . 6 |- ( x = y -> ( x = y \/ x =/= y ) )
10	9, 6	sylibr 133	. . . . 5 |- ( x = y -> DECID x = y )
11	10	a1i 9	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x = y -> DECID x = y ) )
12		ltne 7983	. . . . . . 7 |- ( ( y e. RR /\ y < x ) -> x =/= y )
13	12	ex 114	. . . . . 6 |- ( y e. RR -> ( y < x -> x =/= y ) )
14	13	adantl 275	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( y < x -> x =/= y ) )
15	4, 6	sylibr 133	. . . . 5 |- ( x =/= y -> DECID x = y )
16	14, 15	syl6 33	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( y < x -> DECID x = y ) )
17	8, 11, 16	3jaod 1294	. . 3 |- ( ( x e. RR /\ y e. RR ) -> ( ( x < y \/ x = y \/ y < x ) -> DECID x = y ) )
18	17	ralimdva 2533	. 2 |- ( x e. RR -> ( A. y e. RR ( x < y \/ x = y \/ y < x ) -> A. y e. RR DECID x = y ) )
19	18	ralimia 2527	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 )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 824   \/ w3o 967   e. wcel 2136   =/= wne 2336   A.wral 2444   class class class wbr 3982   RRcr 7752   < clt 7933

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-pre-ltirr 7865

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-pnf 7935  df-mnf 7936  df-ltxr 7938

This theorem is referenced by:  dcapnconstALT  13950