Theorem so 3620

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 1884	. . . . 5 |- x = x
2	1	orci 292	. . . 4 |- (x = x \/ xRx)
3		eleq1 1957	. . . . . . 7 |- (y = x -> (y e. A <-> x e. A))
4	3	anbi2d 678	. . . . . 6 |- (y = x -> ((x e. A /\ y e. A) <-> (x e. A /\ x e. A)))
5		eqeq2 1893	. . . . . . . 8 |- (y = x -> (x = y <-> x = x))
6		breq1 3341	. . . . . . . 8 |- (y = x -> (yRx <-> xRx))
7	5, 6	orbi12d 689	. . . . . . 7 |- (y = x -> ((x = y \/ yRx) <-> (x = x \/ xRx)))
8		breq2 3342	. . . . . . . 8 |- (y = x -> (xRy <-> xRx))
9	8	notbid 673	. . . . . . 7 |- (y = x -> (-. xRy <-> -. xRx))
10	7, 9	bibi12d 691	. . . . . 6 |- (y = x -> (((x = y \/ yRx) <-> -. xRy) <-> ((x = x \/ xRx) <-> -. xRx)))
11	4, 10	imbi12d 688	. . . . 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 1957	. . . . . . . . . 10 |- (z = y -> (z e. A <-> y e. A))
13	12	3anbi3d 1174	. . . . . . . . 9 |- (z = y -> ((x e. A /\ y e. A /\ z e. A) <-> (x e. A /\ y e. A /\ y e. A)))
14	13	imbi1d 675	. . . . . . . 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	simplld 348	. . . . . . . 8 |- ((x e. A /\ y e. A /\ z e. A) -> (xRy <-> -. (x = y \/ yRx)))
17	14, 16	chvarv 1712	. . . . . . 7 |- ((x e. A /\ y e. A /\ y e. A) -> (xRy <-> -. (x = y \/ yRx)))
18	17	3anidm23 1156	. . . . . 6 |- ((x e. A /\ y e. A) -> (xRy <-> -. (x = y \/ yRx)))
19	18	con2bid 585	. . . . 5 |- ((x e. A /\ y e. A) -> ((x = y \/ yRx) <-> -. xRy))
20	11, 19	chvarv 1712	. . . 4 |- ((x e. A /\ x e. A) -> ((x = x \/ xRx) <-> -. xRx))
21	2, 20	mpbii 210	. . 3 |- ((x e. A /\ x e. A) -> -. xRx)
22	21	anidms 480	. 2 |- (x e. A -> -. xRx)
23	15	simprd 352	. 2 |- ((x e. A /\ y e. A /\ z e. A) -> ((xRy /\ yRz) -> xRz))
24	19	biimprd 171	. . 3 |- ((x e. A /\ y e. A) -> (-. xRy -> (x = y \/ yRx)))
25		3orass 861	. . . 4 |- ((xRy \/ x = y \/ yRx) <-> (xRy \/ (x = y \/ yRx)))
26		df-or 241	. . . 4 |- ((xRy \/ (x = y \/ yRx)) <-> (-. xRy -> (x = y \/ yRx)))
27	25, 26	bitri 190	. . 3 |- ((xRy \/ x = y \/ yRx) <-> (-. xRy -> (x = y \/ yRx)))
28	24, 27	sylibr 217	. 2 |- ((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx))
29	22, 23, 28	itlso 3619	1 |- R Or A

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   \/ w3o 857   /\ w3a 858   = wceq 1298   e. wcel 1300   class class class wbr 3338   Or wor 3590

This theorem is referenced by:  ltsopi 6168  ltsopq 6227  ltsosr 6355  ltsor 6413  ltso 6681  xrltso 6729

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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