Theorem tridceq 13590

Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 13577 and redcwlpo 13589). Thus, this is an analytic analogue to lpowlpo 7094. (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 7945	. . . . . . 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 2411	. . . . . 6 |- ( y =/= x <-> x =/= y )
6		dcne 2338	. . . . . 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 7945	. . . . . . 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 1286	. . 3 |- ( ( x e. RR /\ y e. RR ) -> ( ( x < y \/ x = y \/ y < x ) -> DECID x = y ) )
18	17	ralimdva 2524	. 2 |- ( x e. RR -> ( A. y e. RR ( x < y \/ x = y \/ y < x ) -> A. y e. RR DECID x = y ) )
19	18	ralimia 2518	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 820   \/ w3o 962   e. wcel 2128   =/= wne 2327   A.wral 2435   class class class wbr 3965   RRcr 7714   < clt 7895

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-cnex 7806  ax-resscn 7807  ax-pre-ltirr 7827

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4589  df-pnf 7897  df-mnf 7898  df-ltxr 7900

This theorem is referenced by:  dcapnconstALT  13595