Theorem fr3nr 7474

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 7450	. . . . . . 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 486	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝑅 Fr 𝐴)
4		df-tp 4530	. . . . . . 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 4715	. . . . . . . 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 4702	. . . . . . . 8 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐷} ⊆ 𝐴)
10	7, 9	unssd 4113	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ({𝐵, 𝐶} ∪ {𝐷}) ⊆ 𝐴)
11	4, 10	eqsstrid 3963	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐵, 𝐶, 𝐷} ⊆ 𝐴)
12	5	tpnzd 4676	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐵, 𝐶, 𝐷} ≠ ∅)
13		fri 5481	. . . . . 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 5034	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐵))
16	15	notbid 321	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐵))
17	16	ralbidv 3162	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵))
18		breq2 5034	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐶))
19	18	notbid 321	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐶))
20	19	ralbidv 3162	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶))
21		breq2 5034	. . . . . . . . 9 ⊢ (𝑥 = 𝐷 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐷))
22	21	notbid 321	. . . . . . . 8 ⊢ (𝑥 = 𝐷 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐷))
23	22	ralbidv 3162	. . . . . . 7 ⊢ (𝑥 = 𝐷 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷))
24	17, 20, 23	rextpg 4595	. . . . . 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 485	. . . . 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 235	. . . 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 4711	. . . . . . 7 ⊢ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}
28		snssg 4678	. . . . . . . 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 261	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ {𝐵, 𝐶, 𝐷})
31		breq1 5033	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → (𝑦𝑅𝐵 ↔ 𝐷𝑅𝐵))
32	31	notbid 321	. . . . . . 7 ⊢ (𝑦 = 𝐷 → (¬ 𝑦𝑅𝐵 ↔ ¬ 𝐷𝑅𝐵))
33	32	rspcv 3566	. . . . . 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 4709	. . . . . . 7 ⊢ {𝐵} ⊆ {𝐵, 𝐶, 𝐷}
36		snssg 4678	. . . . . . . 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 261	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐵 ∈ {𝐵, 𝐶, 𝐷})
39		breq1 5033	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐶 ↔ 𝐵𝑅𝐶))
40	39	notbid 321	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (¬ 𝑦𝑅𝐶 ↔ ¬ 𝐵𝑅𝐶))
41	40	rspcv 3566	. . . . . 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 4710	. . . . . . 7 ⊢ {𝐶} ⊆ {𝐵, 𝐶, 𝐷}
44		snssg 4678	. . . . . . . 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 261	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ {𝐵, 𝐶, 𝐷})
47		breq1 5033	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝑦𝑅𝐷 ↔ 𝐶𝑅𝐷))
48	47	notbid 321	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (¬ 𝑦𝑅𝐷 ↔ ¬ 𝐶𝑅𝐷))
49	48	rspcv 3566	. . . . . 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 1441	. . . 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 237	. 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 331	1 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  ∅c0 4243  {csn 4525  {cpr 4527  {ctp 4529   class class class wbr 5030   Fr wfr 5475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-fr 5478

This theorem is referenced by:  epne3  7475  dfwe2  7476