Theorem solin 3612

Description: A strict order relation is linear (satisfies trichotomy).

Assertion

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

Proof of Theorem solin

Step	Hyp	Ref	Expression
1		breq1 3341	. . . . 5 |- (x = B -> (xRy <-> BRy))
2		eqeq1 1890	. . . . 5 |- (x = B -> (x = y <-> B = y))
3		breq2 3342	. . . . 5 |- (x = B -> (yRx <-> yRB))
4	1, 2, 3	3orbi123d 1167	. . . 4 |- (x = B -> ((xRy \/ x = y \/ yRx) <-> (BRy \/ B = y \/ yRB)))
5	4	imbi2d 674	. . 3 |- (x = B -> ((R Or A -> (xRy \/ x = y \/ yRx)) <-> (R Or A -> (BRy \/ B = y \/ yRB))))
6		breq2 3342	. . . . 5 |- (y = C -> (BRy <-> BRC))
7		eqeq2 1893	. . . . 5 |- (y = C -> (B = y <-> B = C))
8		breq1 3341	. . . . 5 |- (y = C -> (yRB <-> CRB))
9	6, 7, 8	3orbi123d 1167	. . . 4 |- (y = C -> ((BRy \/ B = y \/ yRB) <-> (BRC \/ B = C \/ CRB)))
10	9	imbi2d 674	. . 3 |- (y = C -> ((R Or A -> (BRy \/ B = y \/ yRB)) <-> (R Or A -> (BRC \/ B = C \/ CRB))))
11		df-so 3604	. . . . 5 |- (R Or A <-> (R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
12		ra42 2157	. . . . . 6 |- (A.x e. A A.y e. A (xRy \/ x = y \/ yRx) -> ((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)))
13	12	adantl 424	. . . . 5 |- ((R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)) -> ((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)))
14	11, 13	sylbi 216	. . . 4 |- (R Or A -> ((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)))
15	14	com12 14	. . 3 |- ((x e. A /\ y e. A) -> (R Or A -> (xRy \/ x = y \/ yRx)))
16	5, 10, 15	vtocl2ga 2353	. 2 |- ((B e. A /\ C e. A) -> (R Or A -> (BRC \/ B = C \/ CRB)))
17	16	impcom 378	1 |- ((R Or A /\ (B e. A /\ C e. A)) -> (BRC \/ B = C \/ CRB))

Syntax hints:   -> wi 3   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300  A.wral 2105   class class class wbr 3338   Po wpo 3589   Or wor 3590

This theorem is referenced by:  sotric 3615  sotrieq 3616  wecmpep 3650  wereu 3654  dfwe2OLD 3862  lttri4 6684  soxp 13950  wfrlem10 13966  slttri 14011

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-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-so 3604

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