Theorem fr3nr 2981

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 1851	. . . . 5 |- y e. V
2	1	tpnz 2508	. . . 4 |- {y, z, x} =/= (/)
3		tpex 2933	. . . . 5 |- {y, z, x} e. V
4	3	frc 2975	. . . 4 |- ((R Fr A /\ {y, z, x} (_ A /\ {y, z, x} =/= (/)) -> E.v e. {y, z, x} ({y, z, x} i^i {w | wRv}) = (/))
5	2, 4	mp3an3 908	. . 3 |- ((R Fr A /\ {y, z, x} (_ A) -> E.v e. {y, z, x} ({y, z, x} i^i {w | wRv}) = (/))
6		3jao 889	. . . . . . . 8 |- (((v = y -> ({y, z, x} i^i {w | wRv}) =/= (/)) /\ (v = z -> ({y, z, x} i^i {w | wRv}) =/= (/)) /\ (v = x -> ({y, z, x} i^i {w | wRv}) =/= (/))) -> ((v = y \/ v = z \/ v = x) -> ({y, z, x} i^i {w | wRv}) =/= (/)))
7		breq2 2673	. . . . . . . . . . . 12 |- (v = y -> (wRv <-> wRy))
8	7	abbidv 1614	. . . . . . . . . . 11 |- (v = y -> {w | wRv} = {w | wRy})
9	8	ineq2d 2261	. . . . . . . . . 10 |- (v = y -> ({y, z, x} i^i {w | wRv}) = ({y, z, x} i^i {w | wRy}))
10	9	neeq1d 1631	. . . . . . . . 9 |- (v = y -> (({y, z, x} i^i {w | wRv}) =/= (/) <-> ({y, z, x} i^i {w | wRy}) =/= (/)))
11		brab1 2710	. . . . . . . . . 10 |- (xRy <-> x e. {w | wRy})
12		visset 1851	. . . . . . . . . . . 12 |- x e. V
13	12	tpi3 2504	. . . . . . . . . . 11 |- x e. {y, z, x}
14		inelcm 2368	. . . . . . . . . . 11 |- ((x e. {y, z, x} /\ x e. {w | wRy}) -> ({y, z, x} i^i {w | wRy}) =/= (/))
15	13, 14	mpan 698	. . . . . . . . . 10 |- (x e. {w | wRy} -> ({y, z, x} i^i {w | wRy}) =/= (/))
16	11, 15	sylbi 197	. . . . . . . . 9 |- (xRy -> ({y, z, x} i^i {w | wRy}) =/= (/))
17	10, 16	syl5cbir 209	. . . . . . . 8 |- (xRy -> (v = y -> ({y, z, x} i^i {w | wRv}) =/= (/)))
18		breq2 2673	. . . . . . . . . . . 12 |- (v = z -> (wRv <-> wRz))
19	18	abbidv 1614	. . . . . . . . . . 11 |- (v = z -> {w | wRv} = {w | wRz})
20	19	ineq2d 2261	. . . . . . . . . 10 |- (v = z -> ({y, z, x} i^i {w | wRv}) = ({y, z, x} i^i {w | wRz}))
21	20	neeq1d 1631	. . . . . . . . 9 |- (v = z -> (({y, z, x} i^i {w | wRv}) =/= (/) <-> ({y, z, x} i^i {w | wRz}) =/= (/)))
22		brab1 2710	. . . . . . . . . 10 |- (yRz <-> y e. {w | wRz})
23	1	tpi1 2502	. . . . . . . . . . 11 |- y e. {y, z, x}
24		inelcm 2368	. . . . . . . . . . 11 |- ((y e. {y, z, x} /\ y e. {w | wRz}) -> ({y, z, x} i^i {w | wRz}) =/= (/))
25	23, 24	mpan 698	. . . . . . . . . 10 |- (y e. {w | wRz} -> ({y, z, x} i^i {w | wRz}) =/= (/))
26	22, 25	sylbi 197	. . . . . . . . 9 |- (yRz -> ({y, z, x} i^i {w | wRz}) =/= (/))
27	21, 26	syl5cbir 209	. . . . . . . 8 |- (yRz -> (v = z -> ({y, z, x} i^i {w | wRv}) =/= (/)))
28		breq2 2673	. . . . . . . . . . . 12 |- (v = x -> (wRv <-> wRx))
29	28	abbidv 1614	. . . . . . . . . . 11 |- (v = x -> {w | wRv} = {w | wRx})
30	29	ineq2d 2261	. . . . . . . . . 10 |- (v = x -> ({y, z, x} i^i {w | wRv}) = ({y, z, x} i^i {w | wRx}))
31	30	neeq1d 1631	. . . . . . . . 9 |- (v = x -> (({y, z, x} i^i {w | wRv}) =/= (/) <-> ({y, z, x} i^i {w | wRx}) =/= (/)))
32		brab1 2710	. . . . . . . . . 10 |- (zRx <-> z e. {w | wRx})
33		visset 1851	. . . . . . . . . . . 12 |- z e. V
34	33	tpi2 2503	. . . . . . . . . . 11 |- z e. {y, z, x}
35		inelcm 2368	. . . . . . . . . . 11 |- ((z e. {y, z, x} /\ z e. {w | wRx}) -> ({y, z, x} i^i {w | wRx}) =/= (/))
36	34, 35	mpan 698	. . . . . . . . . 10 |- (z e. {w | wRx} -> ({y, z, x} i^i {w | wRx}) =/= (/))
37	32, 36	sylbi 197	. . . . . . . . 9 |- (zRx -> ({y, z, x} i^i {w | wRx}) =/= (/))
38	31, 37	syl5cbir 209	. . . . . . . 8 |- (zRx -> (v = x -> ({y, z, x} i^i {w | wRv}) =/= (/)))
39	6, 17, 27, 38	syl3an 871	. . . . . . 7 |- ((xRy /\ yRz /\ zRx) -> ((v = y \/ v = z \/ v = x) -> ({y, z, x} i^i {w | wRv}) =/= (/)))
40		visset 1851	. . . . . . . 8 |- v e. V
41	40	eltp 2484	. . . . . . 7 |- (v e. {y, z, x} <-> (v = y \/ v = z \/ v = x))
42	39, 41	syl5ib 204	. . . . . 6 |- ((xRy /\ yRz /\ zRx) -> (v e. {y, z, x} -> ({y, z, x} i^i {w | wRv}) =/= (/)))
43	42	com12 11	. . . . 5 |- (v e. {y, z, x} -> ((xRy /\ yRz /\ zRx) -> ({y, z, x} i^i {w | wRv}) =/= (/)))
44	43	necon2bd 1652	. . . 4 |- (v e. {y, z, x} -> (({y, z, x} i^i {w | wRv}) = (/) -> -. (xRy /\ yRz /\ zRx)))
45	44	r19.23aiv 1781	. . 3 |- (E.v e. {y, z, x} ({y, z, x} i^i {w | wRv}) = (/) -> -. (xRy /\ yRz /\ zRx))
46	5, 45	syl 10	. 2 |- ((R Fr A /\ {y, z, x} (_ A) -> -. (xRy /\ yRz /\ zRx))
47		3anrot 783	. . 3 |- ((x e. A /\ y e. A /\ z e. A) <-> (y e. A /\ z e. A /\ x e. A))
48	1, 33, 12	tpss 2524	. . 3 |- ((y e. A /\ z e. A /\ x e. A) <-> {y, z, x} (_ A)
49	47, 48	bitri 171	. 2 |- ((x e. A /\ y e. A /\ z e. A) <-> {y, z, x} (_ A)
50	46, 49	sylan2b 454	1 |- ((R Fr A /\ (x e. A /\ y e. A /\ z e. A)) -> -. (xRy /\ yRz /\ zRx))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 221   \/ w3o 777   /\ w3a 778   = wceq 988   e. wcel 990  {cab 1499   =/= wne 1622  E.wrex 1684   i^i cin 2090   (_ wss 2091  (/)c0 2324  {ctp 2459   class class class wbr 2669   Fr wfr 2970

This theorem is referenced by:  epne3 2985  dfwe2 2990

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 779  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-tp 2460  df-op 2461  df-uni 2552  df-br 2670  df-fr 2972

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