Theorem so 2855

Description: Deduce strict ordering from its properties.

Hypothesis

Ref	Expression
so.1	|- ((x e. A /\ y e. A /\ z e. A) -> ((xRy <-> -. (x = y \/ yRx)) /\ ((xRy /\ yRz) -> xRz)))

Assertion

Ref	Expression
so	|- R Or A

Distinct variable groups:   x,y,z,R   x,A,y,z

Proof of Theorem so

Step	Hyp	Ref	Expression
1		eqid 1468	. . . . 5 |- x = x
2	1	orci 270	. . . 4 |- (x = x \/ xRx)
3		eleq1 1526	. . . . . . 7 |- (y = x -> (y e. A <-> x e. A))
4	3	anbi2d 614	. . . . . 6 |- (y = x -> ((x e. A /\ y e. A) <-> (x e. A /\ x e. A)))
5		eqeq2 1476	. . . . . . . 8 |- (y = x -> (x = y <-> x = x))
6		breq1 2612	. . . . . . . 8 |- (y = x -> (yRx <-> xRx))
7	5, 6	orbi12d 625	. . . . . . 7 |- (y = x -> ((x = y \/ yRx) <-> (x = x \/ xRx)))
8		breq2 2613	. . . . . . . 8 |- (y = x -> (xRy <-> xRx))
9	8	negbid 609	. . . . . . 7 |- (y = x -> (-. xRy <-> -. xRx))
10	7, 9	bibi12d 627	. . . . . 6 |- (y = x -> (((x = y \/ yRx) <-> -. xRy) <-> ((x = x \/ xRx) <-> -. xRx)))
11	4, 10	imbi12d 624	. . . . 5 |- (y = x -> (((x e. A /\ y e. A) -> ((x = y \/ yRx) <-> -. xRy)) <-> ((x e. A /\ x e. A) -> ((x = x \/ xRx) <-> -. xRx))))
12		eleq1 1526	. . . . . . . . . 10 |- (z = y -> (z e. A <-> y e. A))
13	12	3anbi3d 896	. . . . . . . . 9 |- (z = y -> ((x e. A /\ y e. A /\ z e. A) <-> (x e. A /\ y e. A /\ y e. A)))
14	13	imbi1d 611	. . . . . . . 8 |- (z = y -> (((x e. A /\ y e. A /\ z e. A) -> (xRy <-> -. (x = y \/ yRx))) <-> ((x e. A /\ y e. A /\ y e. A) -> (xRy <-> -. (x = y \/ yRx)))))
15		so.1	. . . . . . . . 9 |- ((x e. A /\ y e. A /\ z e. A) -> ((xRy <-> -. (x = y \/ yRx)) /\ ((xRy /\ yRz) -> xRz)))
16	15	pm3.26d 321	. . . . . . . 8 |- ((x e. A /\ y e. A /\ z e. A) -> (xRy <-> -. (x = y \/ yRx)))
17	14, 16	chvarv 1322	. . . . . . 7 |- ((x e. A /\ y e. A /\ y e. A) -> (xRy <-> -. (x = y \/ yRx)))
18	17	3anidm23 881	. . . . . 6 |- ((x e. A /\ y e. A) -> (xRy <-> -. (x = y \/ yRx)))
19	18	con2bid 524	. . . . 5 |- ((x e. A /\ y e. A) -> ((x = y \/ yRx) <-> -. xRy))
20	11, 19	chvarv 1322	. . . 4 |- ((x e. A /\ x e. A) -> ((x = x \/ xRx) <-> -. xRx))
21	2, 20	mpbii 193	. . 3 |- ((x e. A /\ x e. A) -> -. xRx)
22	21	anidms 434	. 2 |- (x e. A -> -. xRx)
23	15	pm3.27d 325	. 2 |- ((x e. A /\ y e. A /\ z e. A) -> ((xRy /\ yRz) -> xRz))
24	19	biimprd 154	. . 3 |- ((x e. A /\ y e. A) -> (-. xRy -> (x = y \/ yRx)))
25		3orass 776	. . . 4 |- ((xRy \/ x = y \/ yRx) <-> (xRy \/ (x = y \/ yRx)))
26		df-or 224	. . . 4 |- ((xRy \/ (x = y \/ yRx)) <-> (-. xRy -> (x = y \/ yRx)))
27	25, 26	bitr 173	. . 3 |- ((xRy \/ x = y \/ yRx) <-> (-. xRy -> (x = y \/ yRx)))
28	24, 27	sylibr 200	. 2 |- ((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx))
29	22, 23, 28	itlso 2854	1 |- R Or A

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 772   /\ w3a 773   = wceq 953   e. wcel 955   class class class wbr 2609   Or wor 2830

This theorem is referenced by:  ltsopi 4988  ltsopq 5047  ltsosr 5175  ltsor 5233  ltso 5484  xrltso 5527

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-po 2831  df-so 2841

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