Theorem sotrieq 2149

Description: Trichotomy law for strict order relation.

Assertion

Ref	Expression
sotrieq	|- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C <-> -. (BRC \/ CRB)))

Proof of Theorem sotrieq

Step	Hyp	Ref	Expression
1		breq2 2066	. . . . . . . 8 |- (B = C -> (BRB <-> BRC))
2	1	negbid 463	. . . . . . 7 |- (B = C -> (-. BRB <-> -. BRC))
3		sonr 2143	. . . . . . 7 |- ((R Or A /\ B e. A) -> -. BRB)
4	2, 3	syl5bi 183	. . . . . 6 |- (B = C -> ((R Or A /\ B e. A) -> -. BRC))
5		breq2 2066	. . . . . . . 8 |- (B = C -> (CRB <-> CRC))
6	5	negbid 463	. . . . . . 7 |- (B = C -> (-. CRB <-> -. CRC))
7		sonr 2143	. . . . . . 7 |- ((R Or A /\ C e. A) -> -. CRC)
8	6, 7	syl5bir 184	. . . . . 6 |- (B = C -> ((R Or A /\ C e. A) -> -. CRB))
9	4, 8	anim12d 431	. . . . 5 |- (B = C -> (((R Or A /\ B e. A) /\ (R Or A /\ C e. A)) -> (-. BRC /\ -. CRB)))
10	9	com12 13	. . . 4 |- (((R Or A /\ B e. A) /\ (R Or A /\ C e. A)) -> (B = C -> (-. BRC /\ -. CRB)))
11	10	anandis 394	. . 3 |- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C -> (-. BRC /\ -. CRB)))
12		sotric 2148	. . . . . . . . 9 |- ((R Or A /\ (B e. A /\ C e. A)) -> (BRC <-> -. (B = C \/ CRB)))
13	12	bicon2d 404	. . . . . . . 8 |- ((R Or A /\ (B e. A /\ C e. A)) -> ((B = C \/ CRB) <-> -. BRC))
14	13	biimpar 325	. . . . . . 7 |- (((R Or A /\ (B e. A /\ C e. A)) /\ -. BRC) -> (B = C \/ CRB))
15	14	ord 202	. . . . . 6 |- (((R Or A /\ (B e. A /\ C e. A)) /\ -. BRC) -> (-. B = C -> CRB))
16	15	con1d 85	. . . . 5 |- (((R Or A /\ (B e. A /\ C e. A)) /\ -. BRC) -> (-. CRB -> B = C))
17	16	exp 291	. . . 4 |- ((R Or A /\ (B e. A /\ C e. A)) -> (-. BRC -> (-. CRB -> B = C)))
18	17	imp3a 279	. . 3 |- ((R Or A /\ (B e. A /\ C e. A)) -> ((-. BRC /\ -. CRB) -> B = C))
19	11, 18	impbid 397	. 2 |- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C <-> (-. BRC /\ -. CRB)))
20		ioran 254	. 2 |- (-. (BRC \/ CRB) <-> (-. BRC /\ -. CRB))
21	19, 20	syl6bbr 416	1 |- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C <-> -. (BRC \/ CRB)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092   class class class wbr 2054   Or wor 2059

This theorem is referenced by:  sotrieq2 2150  distrlem4pr 3924  addcanpr 3946  sqgt0sr 4009  lttri2t 4280

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-po 2128  df-so 2138

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