Theorem fr3nr 7778

Description: A well-founded relation has no 3-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 10-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
fr3nr	⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵))

Proof of Theorem fr3nr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tpex 7753	. . . . . . 7 ⊢ {𝐵, 𝐶, 𝐷} ∈ V
2	1	a1i 11	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐵, 𝐶, 𝐷} ∈ V)
3		simpl 481	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝑅 Fr 𝐴)
4		df-tp 4635	. . . . . . 7 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
5		simpr1 1191	. . . . . . . . 9 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
6		simpr2 1192	. . . . . . . . 9 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
7	5, 6	prssd 4828	. . . . . . . 8 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐵, 𝐶} ⊆ 𝐴)
8		simpr3 1193	. . . . . . . . 9 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ 𝐴)
9	8	snssd 4815	. . . . . . . 8 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐷} ⊆ 𝐴)
10	7, 9	unssd 4186	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ({𝐵, 𝐶} ∪ {𝐷}) ⊆ 𝐴)
11	4, 10	eqsstrid 4028	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐵, 𝐶, 𝐷} ⊆ 𝐴)
12	5	tpnzd 4787	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐵, 𝐶, 𝐷} ≠ ∅)
13		fri 5640	. . . . . 6 ⊢ ((({𝐵, 𝐶, 𝐷} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝐵, 𝐶, 𝐷} ⊆ 𝐴 ∧ {𝐵, 𝐶, 𝐷} ≠ ∅)) → ∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥)
14	2, 3, 11, 12, 13	syl22anc 837	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥)
15		breq2 5154	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐵))
16	15	notbid 317	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐵))
17	16	ralbidv 3173	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵))
18		breq2 5154	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐶))
19	18	notbid 317	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐶))
20	19	ralbidv 3173	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶))
21		breq2 5154	. . . . . . . . 9 ⊢ (𝑥 = 𝐷 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐷))
22	21	notbid 317	. . . . . . . 8 ⊢ (𝑥 = 𝐷 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐷))
23	22	ralbidv 3173	. . . . . . 7 ⊢ (𝑥 = 𝐷 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷))
24	17, 20, 23	rextpg 4706	. . . . . 6 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷)))
25	24	adantl 480	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷)))
26	14, 25	mpbid 231	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷))
27		snsstp3 4824	. . . . . . 7 ⊢ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}
28		snssg 4790	. . . . . . . 8 ⊢ (𝐷 ∈ 𝐴 → (𝐷 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
29	8, 28	syl 17	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐷 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
30	27, 29	mpbiri 257	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ {𝐵, 𝐶, 𝐷})
31		breq1 5153	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → (𝑦𝑅𝐵 ↔ 𝐷𝑅𝐵))
32	31	notbid 317	. . . . . . 7 ⊢ (𝑦 = 𝐷 → (¬ 𝑦𝑅𝐵 ↔ ¬ 𝐷𝑅𝐵))
33	32	rspcv 3605	. . . . . 6 ⊢ (𝐷 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 → ¬ 𝐷𝑅𝐵))
34	30, 33	syl 17	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 → ¬ 𝐷𝑅𝐵))
35		snsstp1 4822	. . . . . . 7 ⊢ {𝐵} ⊆ {𝐵, 𝐶, 𝐷}
36		snssg 4790	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐵} ⊆ {𝐵, 𝐶, 𝐷}))
37	5, 36	syl 17	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐵 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐵} ⊆ {𝐵, 𝐶, 𝐷}))
38	35, 37	mpbiri 257	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐵 ∈ {𝐵, 𝐶, 𝐷})
39		breq1 5153	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐶 ↔ 𝐵𝑅𝐶))
40	39	notbid 317	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (¬ 𝑦𝑅𝐶 ↔ ¬ 𝐵𝑅𝐶))
41	40	rspcv 3605	. . . . . 6 ⊢ (𝐵 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 → ¬ 𝐵𝑅𝐶))
42	38, 41	syl 17	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 → ¬ 𝐵𝑅𝐶))
43		snsstp2 4823	. . . . . . 7 ⊢ {𝐶} ⊆ {𝐵, 𝐶, 𝐷}
44		snssg 4790	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → (𝐶 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐶} ⊆ {𝐵, 𝐶, 𝐷}))
45	6, 44	syl 17	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐶} ⊆ {𝐵, 𝐶, 𝐷}))
46	43, 45	mpbiri 257	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ {𝐵, 𝐶, 𝐷})
47		breq1 5153	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝑦𝑅𝐷 ↔ 𝐶𝑅𝐷))
48	47	notbid 317	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (¬ 𝑦𝑅𝐷 ↔ ¬ 𝐶𝑅𝐷))
49	48	rspcv 3605	. . . . . 6 ⊢ (𝐶 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷 → ¬ 𝐶𝑅𝐷))
50	46, 49	syl 17	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷 → ¬ 𝐶𝑅𝐷))
51	34, 42, 50	3orim123d 1440	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷) → (¬ 𝐷𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐷)))
52	26, 51	mpd 15	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ 𝐷𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐷))
53		3ianor 1104	. . 3 ⊢ (¬ (𝐷𝑅𝐵 ∧ 𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) ↔ (¬ 𝐷𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐷))
54	52, 53	sylibr 233	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐷𝑅𝐵 ∧ 𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷))
55		3anrot 1097	. 2 ⊢ ((𝐷𝑅𝐵 ∧ 𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) ↔ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵))
56	54, 55	sylnib 327	1 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ w3o 1083   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2936  ∀wral 3057  ∃wrex 3066  Vcvv 3471   ∪ cun 3945   ⊆ wss 3947  ∅c0 4324  {csn 4630  {cpr 4632  {ctp 4634   class class class wbr 5150   Fr wfr 5632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4911  df-br 5151  df-fr 5635

This theorem is referenced by:  epne3  7779  dfwe2  7780