Theorem solin 2145

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 2065	. . . . . 6 |- (x = B -> (xRy <-> BRy))
2		cleq1 1107	. . . . . 6 |- (x = B -> (x = y <-> B = y))
3		breq2 2066	. . . . . 6 |- (x = B -> (yRx <-> yRB))
4	1, 2, 3	bi3ord 635	. . . . 5 |- (x = B -> ((xRy \/ x = y \/ yRx) <-> (BRy \/ B = y \/ yRB)))
5	4	imbi2d 464	. . . 4 |- (x = B -> ((R Or A -> (xRy \/ x = y \/ yRx)) <-> (R Or A -> (BRy \/ B = y \/ yRB))))
6		breq2 2066	. . . . . 6 |- (y = C -> (BRy <-> BRC))
7		cleq2 1110	. . . . . 6 |- (y = C -> (B = y <-> B = C))
8		breq1 2065	. . . . . 6 |- (y = C -> (yRB <-> CRB))
9	6, 7, 8	bi3ord 635	. . . . 5 |- (y = C -> ((BRy \/ B = y \/ yRB) <-> (BRC \/ B = C \/ CRB)))
10	9	imbi2d 464	. . . 4 |- (y = C -> ((R Or A -> (BRy \/ B = y \/ yRB)) <-> (R Or A -> (BRC \/ B = C \/ CRB))))
11		df-so 2138	. . . . . 6 |- (R Or A <-> (R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
12		ra42 1245	. . . . . . 7 |- (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 305	. . . . . 6 |- ((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 174	. . . . 5 |- (R Or A -> ((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)))
15	14	com12 13	. . . 4 |- ((x e. A /\ y e. A) -> (R Or A -> (xRy \/ x = y \/ yRx)))
16	5, 10, 15	vtocl2ga 1388	. . 3 |- ((B e. A /\ C e. A) -> (R Or A -> (BRC \/ B = C \/ CRB)))
17	16	com12 13	. 2 |- (R Or A -> ((B e. A /\ C e. A) -> (BRC \/ B = C \/ CRB)))
18	17	imp 277	1 |- ((R Or A /\ (B e. A /\ C e. A)) -> (BRC \/ B = C \/ CRB))

Syntax hints:   -> wi 2   /\ wa 196   \/ w3o 580   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201   class class class wbr 2054   Po wpo 2058   Or wor 2059

This theorem is referenced by:  sotric 2148  dfwe2 2187  wecmpep 2193  wereu 2197

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-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-so 2138

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