Theorem dfwe2 3861

Description: Alternate definition of well-ordering. Definition 6.24(2) of [TakeutiZaring] p. 30. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
dfwe2	|- (R We A <-> (R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))

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

Proof of Theorem dfwe2

Step	Hyp	Ref	Expression
1		df-we 3644	. 2 |- (R We A <-> (R Fr A /\ R Or A))
2		simpr 350	. . . . 5 |- ((R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)) -> A.x e. A A.y e. A (xRy \/ x = y \/ yRx))
3		ax-1 4	. . . . . . . . . . . . . . 15 |- (xRz -> ((xRy /\ yRz) -> xRz))
4	3	a1i 8	. . . . . . . . . . . . . 14 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> (xRz -> ((xRy /\ yRz) -> xRz)))
5		breq2 3342	. . . . . . . . . . . . . . . . . 18 |- (x = z -> (yRx <-> yRz))
6	5	anbi2d 678	. . . . . . . . . . . . . . . . 17 |- (x = z -> ((xRy /\ yRx) <-> (xRy /\ yRz)))
7	6	notbid 673	. . . . . . . . . . . . . . . 16 |- (x = z -> (-. (xRy /\ yRx) <-> -. (xRy /\ yRz)))
8		fr2nr 3635	. . . . . . . . . . . . . . . . 17 |- ((R Fr A /\ (x e. A /\ y e. A)) -> -. (xRy /\ yRx))
9	8	3adantr3 1037	. . . . . . . . . . . . . . . 16 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> -. (xRy /\ yRx))
10	7, 9	syl5cbi 226	. . . . . . . . . . . . . . 15 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> (x = z -> -. (xRy /\ yRz)))
11		pm2.21 92	. . . . . . . . . . . . . . 15 |- (-. (xRy /\ yRz) -> ((xRy /\ yRz) -> xRz))
12	10, 11	syl6 25	. . . . . . . . . . . . . 14 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> (x = z -> ((xRy /\ yRz) -> xRz)))
13		fr3nr 3859	. . . . . . . . . . . . . . . . 17 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> -. (xRy /\ yRz /\ zRx))
14		df-3an 860	. . . . . . . . . . . . . . . . . . 19 |- ((xRy /\ yRz /\ zRx) <-> ((xRy /\ yRz) /\ zRx))
15	14	biimpri 169	. . . . . . . . . . . . . . . . . 18 |- (((xRy /\ yRz) /\ zRx) -> (xRy /\ yRz /\ zRx))
16	15	ancoms 484	. . . . . . . . . . . . . . . . 17 |- ((zRx /\ (xRy /\ yRz)) -> (xRy /\ yRz /\ zRx))
17	13, 16	nsyl 131	. . . . . . . . . . . . . . . 16 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> -. (zRx /\ (xRy /\ yRz)))
18	17	pm2.21d 94	. . . . . . . . . . . . . . 15 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> ((zRx /\ (xRy /\ yRz)) -> xRz))
19	18	exp3a 405	. . . . . . . . . . . . . 14 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> (zRx -> ((xRy /\ yRz) -> xRz)))
20	4, 12, 19	3jaod 1161	. . . . . . . . . . . . 13 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> ((xRz \/ x = z \/ zRx) -> ((xRy /\ yRz) -> xRz)))
21		frirr 3634	. . . . . . . . . . . . . 14 |- ((R Fr A /\ x e. A) -> -. xRx)
22	21	3ad2antr1 1041	. . . . . . . . . . . . 13 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> -. xRx)
23	20, 22	jctild 662	. . . . . . . . . . . 12 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> ((xRz \/ x = z \/ zRx) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
24	23	ex 402	. . . . . . . . . . 11 |- (R Fr A -> ((x e. A /\ y e. A /\ z e. A) -> ((xRz \/ x = z \/ zRx) -> (-. xRx /\ ((xRy /\ yRz) -> xRz)))))
25	24	a2d 16	. . . . . . . . . 10 |- (R Fr A -> (((x e. A /\ y e. A /\ z e. A) -> (xRz \/ x = z \/ zRx)) -> ((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz)))))
26	25	alimdv 1668	. . . . . . . . 9 |- (R Fr A -> (A.z((x e. A /\ y e. A /\ z e. A) -> (xRz \/ x = z \/ zRx)) -> A.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz)))))
27	26	2alimdv 1670	. . . . . . . 8 |- (R Fr A -> (A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (xRz \/ x = z \/ zRx)) -> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz)))))
28		r3al 2151	. . . . . . . 8 |- (A.x e. A A.y e. A A.z e. A (xRz \/ x = z \/ zRx) <-> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (xRz \/ x = z \/ zRx)))
29		r3al 2151	. . . . . . . 8 |- (A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz)) <-> A.xA.yA.z((x e. A /\ y e. A /\ z e. A) -> (-. xRx /\ ((xRy /\ yRz) -> xRz))))
30	27, 28, 29	3imtr4g 612	. . . . . . 7 |- (R Fr A -> (A.x e. A A.y e. A A.z e. A (xRz \/ x = z \/ zRx) -> A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz))))
31		ralidm 2973	. . . . . . . . 9 |- (A.y e. A A.y e. A (xRy \/ x = y \/ yRx) <-> A.y e. A (xRy \/ x = y \/ yRx))
32		breq2 3342	. . . . . . . . . . . 12 |- (y = z -> (xRy <-> xRz))
33		equequ2 1495	. . . . . . . . . . . 12 |- (y = z -> (x = y <-> x = z))
34		breq1 3341	. . . . . . . . . . . 12 |- (y = z -> (yRx <-> zRx))
35	32, 33, 34	3orbi123d 1167	. . . . . . . . . . 11 |- (y = z -> ((xRy \/ x = y \/ yRx) <-> (xRz \/ x = z \/ zRx)))
36	35	cbvralv 2280	. . . . . . . . . 10 |- (A.y e. A (xRy \/ x = y \/ yRx) <-> A.z e. A (xRz \/ x = z \/ zRx))
37	36	ralbii 2127	. . . . . . . . 9 |- (A.y e. A A.y e. A (xRy \/ x = y \/ yRx) <-> A.y e. A A.z e. A (xRz \/ x = z \/ zRx))
38	31, 37	bitr3i 192	. . . . . . . 8 |- (A.y e. A (xRy \/ x = y \/ yRx) <-> A.y e. A A.z e. A (xRz \/ x = z \/ zRx))
39	38	ralbii 2127	. . . . . . 7 |- (A.x e. A A.y e. A (xRy \/ x = y \/ yRx) <-> A.x e. A A.y e. A A.z e. A (xRz \/ x = z \/ zRx))
40		df-po 3591	. . . . . . 7 |- (R Po A <-> A.x e. A A.y e. A A.z e. A (-. xRx /\ ((xRy /\ yRz) -> xRz)))
41	30, 39, 40	3imtr4g 612	. . . . . 6 |- (R Fr A -> (A.x e. A A.y e. A (xRy \/ x = y \/ yRx) -> R Po A))
42	41	ancrd 323	. . . . 5 |- (R Fr A -> (A.x e. A A.y e. A (xRy \/ x = y \/ yRx) -> (R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx))))
43	2, 42	impbid2 576	. . . 4 |- (R Fr A -> ((R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)) <-> A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
44		df-so 3604	. . . 4 |- (R Or A <-> (R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
45	43, 44	syl5bb 591	. . 3 |- (R Fr A -> (R Or A <-> A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
46	45	pm5.32i 707	. 2 |- ((R Fr A /\ R Or A) <-> (R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
47	1, 46	bitri 190	1 |- (R We A <-> (R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   \/ w3o 857   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105   class class class wbr 3338   Po wpo 3589   Or wor 3590   Fr wfr 3623   We wwe 3624

This theorem is referenced by:  ordon 3863  weinxp 4059  isowe 4880  f1oweALT 4883  dford2 5711  dfon2 13858

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-13 1311  ax-14 1312  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  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

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-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-po 3591  df-so 3604  df-fr 3625  df-we 3644

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