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Theorem fr3nr 7488
 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 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))

Proof of Theorem fr3nr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tpex 7464 . . . . . . 7 {𝐵, 𝐶, 𝐷} ∈ V
21a1i 11 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → {𝐵, 𝐶, 𝐷} ∈ V)
3 simpl 486 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝑅 Fr 𝐴)
4 df-tp 4555 . . . . . . 7 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
5 simpr1 1191 . . . . . . . . 9 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐵𝐴)
6 simpr2 1192 . . . . . . . . 9 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐶𝐴)
75, 6prssd 4739 . . . . . . . 8 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → {𝐵, 𝐶} ⊆ 𝐴)
8 simpr3 1193 . . . . . . . . 9 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐷𝐴)
98snssd 4726 . . . . . . . 8 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → {𝐷} ⊆ 𝐴)
107, 9unssd 4148 . . . . . . 7 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ({𝐵, 𝐶} ∪ {𝐷}) ⊆ 𝐴)
114, 10eqsstrid 4001 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → {𝐵, 𝐶, 𝐷} ⊆ 𝐴)
125tpnzd 4700 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → {𝐵, 𝐶, 𝐷} ≠ ∅)
13 fri 5504 . . . . . 6 ((({𝐵, 𝐶, 𝐷} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝐵, 𝐶, 𝐷} ⊆ 𝐴 ∧ {𝐵, 𝐶, 𝐷} ≠ ∅)) → ∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥)
142, 3, 11, 12, 13syl22anc 837 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥)
15 breq2 5056 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑦𝑅𝑥𝑦𝑅𝐵))
1615notbid 321 . . . . . . . 8 (𝑥 = 𝐵 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐵))
1716ralbidv 3192 . . . . . . 7 (𝑥 = 𝐵 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵))
18 breq2 5056 . . . . . . . . 9 (𝑥 = 𝐶 → (𝑦𝑅𝑥𝑦𝑅𝐶))
1918notbid 321 . . . . . . . 8 (𝑥 = 𝐶 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐶))
2019ralbidv 3192 . . . . . . 7 (𝑥 = 𝐶 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶))
21 breq2 5056 . . . . . . . . 9 (𝑥 = 𝐷 → (𝑦𝑅𝑥𝑦𝑅𝐷))
2221notbid 321 . . . . . . . 8 (𝑥 = 𝐷 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐷))
2322ralbidv 3192 . . . . . . 7 (𝑥 = 𝐷 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷))
2417, 20, 23rextpg 4620 . . . . . 6 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷)))
2524adantl 485 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷)))
2614, 25mpbid 235 . . . 4 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷))
27 snsstp3 4735 . . . . . . 7 {𝐷} ⊆ {𝐵, 𝐶, 𝐷}
28 snssg 4702 . . . . . . . 8 (𝐷𝐴 → (𝐷 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
298, 28syl 17 . . . . . . 7 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐷 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
3027, 29mpbiri 261 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐷 ∈ {𝐵, 𝐶, 𝐷})
31 breq1 5055 . . . . . . . 8 (𝑦 = 𝐷 → (𝑦𝑅𝐵𝐷𝑅𝐵))
3231notbid 321 . . . . . . 7 (𝑦 = 𝐷 → (¬ 𝑦𝑅𝐵 ↔ ¬ 𝐷𝑅𝐵))
3332rspcv 3604 . . . . . 6 (𝐷 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 → ¬ 𝐷𝑅𝐵))
3430, 33syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 → ¬ 𝐷𝑅𝐵))
35 snsstp1 4733 . . . . . . 7 {𝐵} ⊆ {𝐵, 𝐶, 𝐷}
36 snssg 4702 . . . . . . . 8 (𝐵𝐴 → (𝐵 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐵} ⊆ {𝐵, 𝐶, 𝐷}))
375, 36syl 17 . . . . . . 7 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐵 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐵} ⊆ {𝐵, 𝐶, 𝐷}))
3835, 37mpbiri 261 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐵 ∈ {𝐵, 𝐶, 𝐷})
39 breq1 5055 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑅𝐶𝐵𝑅𝐶))
4039notbid 321 . . . . . . 7 (𝑦 = 𝐵 → (¬ 𝑦𝑅𝐶 ↔ ¬ 𝐵𝑅𝐶))
4140rspcv 3604 . . . . . 6 (𝐵 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 → ¬ 𝐵𝑅𝐶))
4238, 41syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 → ¬ 𝐵𝑅𝐶))
43 snsstp2 4734 . . . . . . 7 {𝐶} ⊆ {𝐵, 𝐶, 𝐷}
44 snssg 4702 . . . . . . . 8 (𝐶𝐴 → (𝐶 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐶} ⊆ {𝐵, 𝐶, 𝐷}))
456, 44syl 17 . . . . . . 7 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐶 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐶} ⊆ {𝐵, 𝐶, 𝐷}))
4643, 45mpbiri 261 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐶 ∈ {𝐵, 𝐶, 𝐷})
47 breq1 5055 . . . . . . . 8 (𝑦 = 𝐶 → (𝑦𝑅𝐷𝐶𝑅𝐷))
4847notbid 321 . . . . . . 7 (𝑦 = 𝐶 → (¬ 𝑦𝑅𝐷 ↔ ¬ 𝐶𝑅𝐷))
4948rspcv 3604 . . . . . 6 (𝐶 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷 → ¬ 𝐶𝑅𝐷))
5046, 49syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷 → ¬ 𝐶𝑅𝐷))
5134, 42, 503orim123d 1441 . . . 4 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷) → (¬ 𝐷𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐷)))
5226, 51mpd 15 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (¬ 𝐷𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐷))
53 3ianor 1104 . . 3 (¬ (𝐷𝑅𝐵𝐵𝑅𝐶𝐶𝑅𝐷) ↔ (¬ 𝐷𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐷))
5452, 53sylibr 237 . 2 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐷𝑅𝐵𝐵𝑅𝐶𝐶𝑅𝐷))
55 3anrot 1097 . 2 ((𝐷𝑅𝐵𝐵𝑅𝐶𝐶𝑅𝐷) ↔ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))
5654, 55sylnib 331 1 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  ∃wrex 3134  Vcvv 3480   ∪ cun 3917   ⊆ wss 3919  ∅c0 4276  {csn 4550  {cpr 4552  {ctp 4554   class class class wbr 5052   Fr wfr 5498 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455 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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-fr 5501 This theorem is referenced by:  epne3  7489  dfwe2  7490
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