Theorem sotrieqOLD 3617

Description: Trichotomy law for strict order relation.

Assertion

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

Proof of Theorem sotrieqOLD

Step	Hyp	Ref	Expression
1		breq2 3342	. . . . . . . 8 |- (B = C -> (BRB <-> BRC))
2	1	notbid 673	. . . . . . 7 |- (B = C -> (-. BRB <-> -. BRC))
3		sonr 3610	. . . . . . 7 |- ((R Or A /\ B e. A) -> -. BRB)
4	2, 3	syl5bi 225	. . . . . 6 |- (B = C -> ((R Or A /\ B e. A) -> -. BRC))
5		breq2 3342	. . . . . . . 8 |- (B = C -> (CRB <-> CRC))
6	5	notbid 673	. . . . . . 7 |- (B = C -> (-. CRB <-> -. CRC))
7		sonr 3610	. . . . . . 7 |- ((R Or A /\ C e. A) -> -. CRC)
8	6, 7	syl5bir 227	. . . . . 6 |- (B = C -> ((R Or A /\ C e. A) -> -. CRB))
9	4, 8	anim12d 617	. . . . 5 |- (B = C -> (((R Or A /\ B e. A) /\ (R Or A /\ C e. A)) -> (-. BRC /\ -. CRB)))
10	9	com12 14	. . . 4 |- (((R Or A /\ B e. A) /\ (R Or A /\ C e. A)) -> (B = C -> (-. BRC /\ -. CRB)))
11	10	anandis 570	. . 3 |- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C -> (-. BRC /\ -. CRB)))
12		sotric 3615	. . . . . . . 8 |- ((R Or A /\ (B e. A /\ C e. A)) -> (BRC <-> -. (B = C \/ CRB)))
13	12	con2bid 585	. . . . . . 7 |- ((R Or A /\ (B e. A /\ C e. A)) -> ((B = C \/ CRB) <-> -. BRC))
14	13	biimpar 461	. . . . . 6 |- (((R Or A /\ (B e. A /\ C e. A)) /\ -. BRC) -> (B = C \/ CRB))
15	14	ord 249	. . . . 5 |- (((R Or A /\ (B e. A /\ C e. A)) /\ -. BRC) -> (-. B = C -> CRB))
16	15	con1d 109	. . . 4 |- (((R Or A /\ (B e. A /\ C e. A)) /\ -. BRC) -> (-. CRB -> B = C))
17	16	expimpd 404	. . 3 |- ((R Or A /\ (B e. A /\ C e. A)) -> ((-. BRC /\ -. CRB) -> B = C))
18	11, 17	impbid 574	. 2 |- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C <-> (-. BRC /\ -. CRB)))
19		ioran 331	. 2 |- (-. (BRC \/ CRB) <-> (-. BRC /\ -. CRB))
20	18, 19	syl6bbr 597	1 |- ((R Or A /\ (B e. A /\ C e. A)) -> (B = C <-> -. (BRC \/ CRB)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   class class class wbr 3338   Or wor 3590

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-po 3591  df-so 3604

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