Theorem solin 2857

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 2622	. . . . 5 |- (x = B -> (xRy <-> BRy))
2		eqeq1 1481	. . . . 5 |- (x = B -> (x = y <-> B = y))
3		breq2 2623	. . . . 5 |- (x = B -> (yRx <-> yRB))
4	1, 2, 3	3orbi123d 892	. . . 4 |- (x = B -> ((xRy \/ x = y \/ yRx) <-> (BRy \/ B = y \/ yRB)))
5	4	imbi2d 612	. . 3 |- (x = B -> ((R Or A -> (xRy \/ x = y \/ yRx)) <-> (R Or A -> (BRy \/ B = y \/ yRB))))
6		breq2 2623	. . . . 5 |- (y = C -> (BRy <-> BRC))
7		eqeq2 1484	. . . . 5 |- (y = C -> (B = y <-> B = C))
8		breq1 2622	. . . . 5 |- (y = C -> (yRB <-> CRB))
9	6, 7, 8	3orbi123d 892	. . . 4 |- (y = C -> ((BRy \/ B = y \/ yRB) <-> (BRC \/ B = C \/ CRB)))
10	9	imbi2d 612	. . 3 |- (y = C -> ((R Or A -> (BRy \/ B = y \/ yRB)) <-> (R Or A -> (BRC \/ B = C \/ CRB))))
11		df-so 2850	. . . . 5 |- (R Or A <-> (R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
12		ra42 1696	. . . . . 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 388	. . . . 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 199	. . . 4 |- (R Or A -> ((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)))
15	14	com12 11	. . 3 |- ((x e. A /\ y e. A) -> (R Or A -> (xRy \/ x = y \/ yRx)))
16	5, 10, 15	vtocl2ga 1853	. 2 |- ((B e. A /\ C e. A) -> (R Or A -> (BRC \/ B = C \/ CRB)))
17	16	impcom 351	1 |- ((R Or A /\ (B e. A /\ C e. A)) -> (BRC \/ B = C \/ CRB))

Syntax hints:   -> wi 3   /\ wa 223   \/ w3o 774   = wceq 956   e. wcel 958  A.wral 1645   class class class wbr 2619   Po wpo 2838   Or wor 2839

This theorem is referenced by:  sotric 2860  dfwe2 2935  wecmpep 2941  wereu 2945

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-so 2850

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