Theorem tridc 6969

Description: A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.)

Hypotheses

Ref	Expression
tridc.po	|- ( ph -> R Po A )
tridc.tri	|- ( ph -> A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) )
tridc.b	|- ( ph -> B e. A )
tridc.c	|- ( ph -> C e. A )

Assertion

Ref	Expression
tridc	|- ( ph -> DECID B R C )

Distinct variable groups:   x, A, y   x, B, y   y, C   x, R, y

Allowed substitution hints:   ph( x, y)   C( x)

Proof of Theorem tridc

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 |- ( ( ph /\ B R C ) -> B R C )
2	1	orcd 734	. . 3 |- ( ( ph /\ B R C ) -> ( B R C \/ -. B R C ) )
3		df-dc 836	. . 3 |- (DECID B R C <-> ( B R C \/ -. B R C ) )
4	2, 3	sylibr 134	. 2 |- ( ( ph /\ B R C ) -> DECID B R C )
5		tridc.po	. . . . . . 7 |- ( ph -> R Po A )
6		tridc.c	. . . . . . 7 |- ( ph -> C e. A )
7		poirr 4343	. . . . . . 7 |- ( ( R Po A /\ C e. A ) -> -. C R C )
8	5, 6, 7	syl2anc 411	. . . . . 6 |- ( ph -> -. C R C )
9	8	adantr 276	. . . . 5 |- ( ( ph /\ B = C ) -> -. C R C )
10		simpr 110	. . . . . 6 |- ( ( ph /\ B = C ) -> B = C )
11	10	breq1d 4044	. . . . 5 |- ( ( ph /\ B = C ) -> ( B R C <-> C R C ) )
12	9, 11	mtbird 674	. . . 4 |- ( ( ph /\ B = C ) -> -. B R C )
13	12	olcd 735	. . 3 |- ( ( ph /\ B = C ) -> ( B R C \/ -. B R C ) )
14	13, 3	sylibr 134	. 2 |- ( ( ph /\ B = C ) -> DECID B R C )
15		tridc.b	. . . . . . 7 |- ( ph -> B e. A )
16		po2nr 4345	. . . . . . 7 |- ( ( R Po A /\ ( C e. A /\ B e. A ) ) -> -. ( C R B /\ B R C ) )
17	5, 6, 15, 16	syl12anc 1247	. . . . . 6 |- ( ph -> -. ( C R B /\ B R C ) )
18	17	adantr 276	. . . . 5 |- ( ( ph /\ C R B ) -> -. ( C R B /\ B R C ) )
19		simplr 528	. . . . . 6 |- ( ( ( ph /\ C R B ) /\ B R C ) -> C R B )
20		simpr 110	. . . . . 6 |- ( ( ( ph /\ C R B ) /\ B R C ) -> B R C )
21	19, 20	jca 306	. . . . 5 |- ( ( ( ph /\ C R B ) /\ B R C ) -> ( C R B /\ B R C ) )
22	18, 21	mtand 666	. . . 4 |- ( ( ph /\ C R B ) -> -. B R C )
23	22	olcd 735	. . 3 |- ( ( ph /\ C R B ) -> ( B R C \/ -. B R C ) )
24	23, 3	sylibr 134	. 2 |- ( ( ph /\ C R B ) -> DECID B R C )
25		tridc.tri	. . 3 |- ( ph -> A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) )
26		breq1 4037	. . . . 5 |- ( x = B -> ( x R y <-> B R y ) )
27		eqeq1 2203	. . . . 5 |- ( x = B -> ( x = y <-> B = y ) )
28		breq2 4038	. . . . 5 |- ( x = B -> ( y R x <-> y R B ) )
29	26, 27, 28	3orbi123d 1322	. . . 4 |- ( x = B -> ( ( x R y \/ x = y \/ y R x ) <-> ( B R y \/ B = y \/ y R B ) ) )
30		breq2 4038	. . . . 5 |- ( y = C -> ( B R y <-> B R C ) )
31		eqeq2 2206	. . . . 5 |- ( y = C -> ( B = y <-> B = C ) )
32		breq1 4037	. . . . 5 |- ( y = C -> ( y R B <-> C R B ) )
33	30, 31, 32	3orbi123d 1322	. . . 4 |- ( y = C -> ( ( B R y \/ B = y \/ y R B ) <-> ( B R C \/ B = C \/ C R B ) ) )
34	29, 33	rspc2va 2882	. . 3 |- ( ( ( B e. A /\ C e. A ) /\ A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) ) -> ( B R C \/ B = C \/ C R B ) )
35	15, 6, 25, 34	syl21anc 1248	. 2 |- ( ph -> ( B R C \/ B = C \/ C R B ) )
36	4, 14, 24, 35	mpjao3dan 1318	1 |- ( ph -> DECID B R C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   \/ w3o 979   = wceq 1364   e. wcel 2167   A.wral 2475   class class class wbr 4034   Po wpo 4330

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-po 4332

This theorem is referenced by:  fimax2gtrilemstep  6970