Theorem tridceq 14774

Description: Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 14761 and redcwlpo 14773). Thus, this is an analytic analogue to lpowlpo 7165. (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 8041	. . . . . . 7 |- ( ( x e. RR /\ x < y ) -> y =/= x )
2	1	ex 115	. . . . . 6 |- ( x e. RR -> ( x < y -> y =/= x ) )
3	2	adantr 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 ) )
7	4, 5, 6	3imtr4i 201	. . . . 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 712	. . . . . 6 |- ( x = y -> ( x = y \/ x =/= y ) )
10	9, 6	sylibr 134	. . . . 5 |- ( x = y -> DECID x = y )
11	10	a1i 9	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( x = y -> DECID x = y ) )
12		ltne 8041	. . . . . . 7 |- ( ( y e. RR /\ y < x ) -> x =/= y )
13	12	ex 115	. . . . . 6 |- ( y e. RR -> ( y < x -> x =/= y ) )
14	13	adantl 277	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( y < x -> x =/= y ) )
15	4, 6	sylibr 134	. . . . 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 1304	. . 3 |- ( ( x e. RR /\ y e. RR ) -> ( ( x < y \/ x = y \/ y < x ) -> DECID x = y ) )
18	17	ralimdva 2544	. 2 |- ( x e. RR -> ( A. y e. RR ( x < y \/ x = y \/ y < x ) -> A. y e. RR DECID x = y ) )
19	18	ralimia 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 )

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 4003   RRcr 7809   < clt 7991

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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltirr 7922

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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-pnf 7993  df-mnf 7994  df-ltxr 7996

This theorem is referenced by:  dcapnconstALT  14779