Theorem fr3nr 6979

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 6957	. . . . . . 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 473	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝑅 Fr 𝐴)
4		df-tp 4182	. . . . . . 7 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
5		simpr1 1067	. . . . . . . . 9 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
6		simpr2 1068	. . . . . . . . 9 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
7		prssi 4353	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
8	5, 6, 7	syl2anc 693	. . . . . . . 8 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐵, 𝐶} ⊆ 𝐴)
9		simpr3 1069	. . . . . . . . 9 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ 𝐴)
10	9	snssd 4340	. . . . . . . 8 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐷} ⊆ 𝐴)
11	8, 10	unssd 3789	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ({𝐵, 𝐶} ∪ {𝐷}) ⊆ 𝐴)
12	4, 11	syl5eqss 3649	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐵, 𝐶, 𝐷} ⊆ 𝐴)
13	5	tpnzd 4314	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → {𝐵, 𝐶, 𝐷} ≠ ∅)
14		fri 5076	. . . . . 6 ⊢ ((({𝐵, 𝐶, 𝐷} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝐵, 𝐶, 𝐷} ⊆ 𝐴 ∧ {𝐵, 𝐶, 𝐷} ≠ ∅)) → ∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥)
15	2, 3, 12, 13, 14	syl22anc 1327	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥)
16		breq2 4657	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐵))
17	16	notbid 308	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐵))
18	17	ralbidv 2986	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵))
19		breq2 4657	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐶))
20	19	notbid 308	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐶))
21	20	ralbidv 2986	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶))
22		breq2 4657	. . . . . . . . 9 ⊢ (𝑥 = 𝐷 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐷))
23	22	notbid 308	. . . . . . . 8 ⊢ (𝑥 = 𝐷 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐷))
24	23	ralbidv 2986	. . . . . . 7 ⊢ (𝑥 = 𝐷 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷))
25	18, 21, 24	rextpg 4237	. . . . . 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)))
26	25	adantl 482	. . . . 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)))
27	15, 26	mpbid 222	. . . 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))
28		snsstp3 4349	. . . . . . 7 ⊢ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}
29		snssg 4327	. . . . . . . 8 ⊢ (𝐷 ∈ 𝐴 → (𝐷 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
30	9, 29	syl 17	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐷 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
31	28, 30	mpbiri 248	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐷 ∈ {𝐵, 𝐶, 𝐷})
32		breq1 4656	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → (𝑦𝑅𝐵 ↔ 𝐷𝑅𝐵))
33	32	notbid 308	. . . . . . 7 ⊢ (𝑦 = 𝐷 → (¬ 𝑦𝑅𝐵 ↔ ¬ 𝐷𝑅𝐵))
34	33	rspcv 3305	. . . . . 6 ⊢ (𝐷 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 → ¬ 𝐷𝑅𝐵))
35	31, 34	syl 17	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 → ¬ 𝐷𝑅𝐵))
36		snsstp1 4347	. . . . . . 7 ⊢ {𝐵} ⊆ {𝐵, 𝐶, 𝐷}
37		snssg 4327	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐵} ⊆ {𝐵, 𝐶, 𝐷}))
38	5, 37	syl 17	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐵 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐵} ⊆ {𝐵, 𝐶, 𝐷}))
39	36, 38	mpbiri 248	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐵 ∈ {𝐵, 𝐶, 𝐷})
40		breq1 4656	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐶 ↔ 𝐵𝑅𝐶))
41	40	notbid 308	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (¬ 𝑦𝑅𝐶 ↔ ¬ 𝐵𝑅𝐶))
42	41	rspcv 3305	. . . . . 6 ⊢ (𝐵 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 → ¬ 𝐵𝑅𝐶))
43	39, 42	syl 17	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 → ¬ 𝐵𝑅𝐶))
44		snsstp2 4348	. . . . . . 7 ⊢ {𝐶} ⊆ {𝐵, 𝐶, 𝐷}
45		snssg 4327	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → (𝐶 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐶} ⊆ {𝐵, 𝐶, 𝐷}))
46	6, 45	syl 17	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐶} ⊆ {𝐵, 𝐶, 𝐷}))
47	44, 46	mpbiri 248	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → 𝐶 ∈ {𝐵, 𝐶, 𝐷})
48		breq1 4656	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝑦𝑅𝐷 ↔ 𝐶𝑅𝐷))
49	48	notbid 308	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (¬ 𝑦𝑅𝐷 ↔ ¬ 𝐶𝑅𝐷))
50	49	rspcv 3305	. . . . . 6 ⊢ (𝐶 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷 → ¬ 𝐶𝑅𝐷))
51	47, 50	syl 17	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷 → ¬ 𝐶𝑅𝐷))
52	35, 43, 51	3orim123d 1407	. . . 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)))
53	27, 52	mpd 15	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ 𝐷𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐷))
54		3ianor 1055	. . 3 ⊢ (¬ (𝐷𝑅𝐵 ∧ 𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) ↔ (¬ 𝐷𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐷))
55	53, 54	sylibr 224	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐷𝑅𝐵 ∧ 𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷))
56		3anrot 1043	. 2 ⊢ ((𝐷𝑅𝐵 ∧ 𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) ↔ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵))
57	55, 56	sylnib 318	1 ⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  {ctp 4181   class class class wbr 4653   Fr wfr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-fr 5073

This theorem is referenced by:  epne3  6980  dfwe2  6981