Theorem fr3nr 3859

Description: A founded relation has no 3-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30.

Assertion

Ref	Expression
fr3nr	|- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> -. (xRy /\ yRz /\ zRx))

Proof of Theorem fr3nr

Step	Hyp	Ref	Expression
1		visset 2295	. . . . 5 |- y e. _V
2	1	tpnz 3121	. . . 4 |- {y, z, x} =/= (/)
3		tpex 3804	. . . . 5 |- {y, z, x} e. _V
4	3	frc 3629	. . . 4 |- ((R Fr A /\ {y, z, x} C_ A /\ {y, z, x} =/= (/)) -> E.v e. {y, z, x} ({y, z, x} i^i {w | wRv}) = (/))
5	2, 4	mp3an3 1180	. . 3 |- ((R Fr A /\ {y, z, x} C_ A) -> E.v e. {y, z, x} ({y, z, x} i^i {w | wRv}) = (/))
6		breq2 3342	. . . . . . . . . . . 12 |- (v = y -> (wRv <-> wRy))
7	6	abbidv 2008	. . . . . . . . . . 11 |- (v = y -> {w | wRv} = {w | wRy})
8	7	ineq2d 2796	. . . . . . . . . 10 |- (v = y -> ({y, z, x} i^i {w | wRv}) = ({y, z, x} i^i {w | wRy}))
9	8	neeq1d 2028	. . . . . . . . 9 |- (v = y -> (({y, z, x} i^i {w | wRv}) =/= (/) <-> ({y, z, x} i^i {w | wRy}) =/= (/)))
10		brab1 3384	. . . . . . . . . 10 |- (xRy <-> x e. {w | wRy})
11		visset 2295	. . . . . . . . . . . 12 |- x e. _V
12	11	tpid3 3116	. . . . . . . . . . 11 |- x e. {y, z, x}
13		inelcm 2928	. . . . . . . . . . 11 |- ((x e. {y, z, x} /\ x e. {w | wRy}) -> ({y, z, x} i^i {w | wRy}) =/= (/))
14	12, 13	mpan 759	. . . . . . . . . 10 |- (x e. {w | wRy} -> ({y, z, x} i^i {w | wRy}) =/= (/))
15	10, 14	sylbi 216	. . . . . . . . 9 |- (xRy -> ({y, z, x} i^i {w | wRy}) =/= (/))
16	9, 15	syl5cbir 228	. . . . . . . 8 |- (xRy -> (v = y -> ({y, z, x} i^i {w | wRv}) =/= (/)))
17		breq2 3342	. . . . . . . . . . . 12 |- (v = z -> (wRv <-> wRz))
18	17	abbidv 2008	. . . . . . . . . . 11 |- (v = z -> {w | wRv} = {w | wRz})
19	18	ineq2d 2796	. . . . . . . . . 10 |- (v = z -> ({y, z, x} i^i {w | wRv}) = ({y, z, x} i^i {w | wRz}))
20	19	neeq1d 2028	. . . . . . . . 9 |- (v = z -> (({y, z, x} i^i {w | wRv}) =/= (/) <-> ({y, z, x} i^i {w | wRz}) =/= (/)))
21		brab1 3384	. . . . . . . . . 10 |- (yRz <-> y e. {w | wRz})
22	1	tpid1 3111	. . . . . . . . . . 11 |- y e. {y, z, x}
23		inelcm 2928	. . . . . . . . . . 11 |- ((y e. {y, z, x} /\ y e. {w | wRz}) -> ({y, z, x} i^i {w | wRz}) =/= (/))
24	22, 23	mpan 759	. . . . . . . . . 10 |- (y e. {w | wRz} -> ({y, z, x} i^i {w | wRz}) =/= (/))
25	21, 24	sylbi 216	. . . . . . . . 9 |- (yRz -> ({y, z, x} i^i {w | wRz}) =/= (/))
26	20, 25	syl5cbir 228	. . . . . . . 8 |- (yRz -> (v = z -> ({y, z, x} i^i {w | wRv}) =/= (/)))
27		breq2 3342	. . . . . . . . . . . 12 |- (v = x -> (wRv <-> wRx))
28	27	abbidv 2008	. . . . . . . . . . 11 |- (v = x -> {w | wRv} = {w | wRx})
29	28	ineq2d 2796	. . . . . . . . . 10 |- (v = x -> ({y, z, x} i^i {w | wRv}) = ({y, z, x} i^i {w | wRx}))
30	29	neeq1d 2028	. . . . . . . . 9 |- (v = x -> (({y, z, x} i^i {w | wRv}) =/= (/) <-> ({y, z, x} i^i {w | wRx}) =/= (/)))
31		brab1 3384	. . . . . . . . . 10 |- (zRx <-> z e. {w | wRx})
32		visset 2295	. . . . . . . . . . . 12 |- z e. _V
33	32	tpid2 3113	. . . . . . . . . . 11 |- z e. {y, z, x}
34		inelcm 2928	. . . . . . . . . . 11 |- ((z e. {y, z, x} /\ z e. {w | wRx}) -> ({y, z, x} i^i {w | wRx}) =/= (/))
35	33, 34	mpan 759	. . . . . . . . . 10 |- (z e. {w | wRx} -> ({y, z, x} i^i {w | wRx}) =/= (/))
36	31, 35	sylbi 216	. . . . . . . . 9 |- (zRx -> ({y, z, x} i^i {w | wRx}) =/= (/))
37	30, 36	syl5cbir 228	. . . . . . . 8 |- (zRx -> (v = x -> ({y, z, x} i^i {w | wRv}) =/= (/)))
38	16, 26, 37	3jaao 1164	. . . . . . 7 |- ((xRy /\ yRz /\ zRx) -> ((v = y \/ v = z \/ v = x) -> ({y, z, x} i^i {w | wRv}) =/= (/)))
39		visset 2295	. . . . . . . 8 |- v e. _V
40	39	eltp 3074	. . . . . . 7 |- (v e. {y, z, x} <-> (v = y \/ v = z \/ v = x))
41	38, 40	syl5ib 223	. . . . . 6 |- ((xRy /\ yRz /\ zRx) -> (v e. {y, z, x} -> ({y, z, x} i^i {w | wRv}) =/= (/)))
42	41	com12 14	. . . . 5 |- (v e. {y, z, x} -> ((xRy /\ yRz /\ zRx) -> ({y, z, x} i^i {w | wRv}) =/= (/)))
43	42	necon2bd 2057	. . . 4 |- (v e. {y, z, x} -> (({y, z, x} i^i {w | wRv}) = (/) -> -. (xRy /\ yRz /\ zRx)))
44	43	r19.23aiv 2211	. . 3 |- (E.v e. {y, z, x} ({y, z, x} i^i {w | wRv}) = (/) -> -. (xRy /\ yRz /\ zRx))
45	5, 44	syl 12	. 2 |- ((R Fr A /\ {y, z, x} C_ A) -> -. (xRy /\ yRz /\ zRx))
46		3anrot 863	. . 3 |- ((x e. A /\ y e. A /\ z e. A) <-> (y e. A /\ z e. A /\ x e. A))
47	1, 32, 11	tpss 3145	. . 3 |- ((y e. A /\ z e. A /\ x e. A) <-> {y, z, x} C_ A)
48	46, 47	bitri 190	. 2 |- ((x e. A /\ y e. A /\ z e. A) <-> {y, z, x} C_ A)
49	45, 48	sylan2b 501	1 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> -. (xRy /\ yRz /\ zRx))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   \/ w3o 857   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875  {ctp 3051   class class class wbr 3338   Fr wfr 3623

This theorem is referenced by:  epne3 3860  dfwe2 3861  dfwe2OLD 3862

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-fr 3625

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