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Theorem fr3nr 7136
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 7114 . . . . . . 7 {𝐵, 𝐶, 𝐷} ∈ V
21a1i 11 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → {𝐵, 𝐶, 𝐷} ∈ V)
3 simpl 474 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝑅 Fr 𝐴)
4 df-tp 4318 . . . . . . 7 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
5 simpr1 1231 . . . . . . . . 9 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐵𝐴)
6 simpr2 1233 . . . . . . . . 9 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐶𝐴)
7 prssi 4490 . . . . . . . . 9 ((𝐵𝐴𝐶𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
85, 6, 7syl2anc 696 . . . . . . . 8 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → {𝐵, 𝐶} ⊆ 𝐴)
9 simpr3 1235 . . . . . . . . 9 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐷𝐴)
109snssd 4477 . . . . . . . 8 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → {𝐷} ⊆ 𝐴)
118, 10unssd 3924 . . . . . . 7 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ({𝐵, 𝐶} ∪ {𝐷}) ⊆ 𝐴)
124, 11syl5eqss 3782 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → {𝐵, 𝐶, 𝐷} ⊆ 𝐴)
135tpnzd 4449 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → {𝐵, 𝐶, 𝐷} ≠ ∅)
14 fri 5220 . . . . . 6 ((({𝐵, 𝐶, 𝐷} ∈ V ∧ 𝑅 Fr 𝐴) ∧ ({𝐵, 𝐶, 𝐷} ⊆ 𝐴 ∧ {𝐵, 𝐶, 𝐷} ≠ ∅)) → ∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥)
152, 3, 12, 13, 14syl22anc 1474 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ∃𝑥 ∈ {𝐵, 𝐶, 𝐷}∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥)
16 breq2 4800 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑦𝑅𝑥𝑦𝑅𝐵))
1716notbid 307 . . . . . . . 8 (𝑥 = 𝐵 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐵))
1817ralbidv 3116 . . . . . . 7 (𝑥 = 𝐵 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵))
19 breq2 4800 . . . . . . . . 9 (𝑥 = 𝐶 → (𝑦𝑅𝑥𝑦𝑅𝐶))
2019notbid 307 . . . . . . . 8 (𝑥 = 𝐶 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐶))
2120ralbidv 3116 . . . . . . 7 (𝑥 = 𝐶 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶))
22 breq2 4800 . . . . . . . . 9 (𝑥 = 𝐷 → (𝑦𝑅𝑥𝑦𝑅𝐷))
2322notbid 307 . . . . . . . 8 (𝑥 = 𝐷 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝐷))
2423ralbidv 3116 . . . . . . 7 (𝑥 = 𝐷 → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷))
2518, 21, 24rextpg 4373 . . . . . 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)))
2625adantl 473 . . . . 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)))
2715, 26mpbid 222 . . . 4 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷))
28 snsstp3 4486 . . . . . . 7 {𝐷} ⊆ {𝐵, 𝐶, 𝐷}
29 snssg 4451 . . . . . . . 8 (𝐷𝐴 → (𝐷 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
309, 29syl 17 . . . . . . 7 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐷 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
3128, 30mpbiri 248 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐷 ∈ {𝐵, 𝐶, 𝐷})
32 breq1 4799 . . . . . . . 8 (𝑦 = 𝐷 → (𝑦𝑅𝐵𝐷𝑅𝐵))
3332notbid 307 . . . . . . 7 (𝑦 = 𝐷 → (¬ 𝑦𝑅𝐵 ↔ ¬ 𝐷𝑅𝐵))
3433rspcv 3437 . . . . . 6 (𝐷 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 → ¬ 𝐷𝑅𝐵))
3531, 34syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 → ¬ 𝐷𝑅𝐵))
36 snsstp1 4484 . . . . . . 7 {𝐵} ⊆ {𝐵, 𝐶, 𝐷}
37 snssg 4451 . . . . . . . 8 (𝐵𝐴 → (𝐵 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐵} ⊆ {𝐵, 𝐶, 𝐷}))
385, 37syl 17 . . . . . . 7 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐵 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐵} ⊆ {𝐵, 𝐶, 𝐷}))
3936, 38mpbiri 248 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐵 ∈ {𝐵, 𝐶, 𝐷})
40 breq1 4799 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑅𝐶𝐵𝑅𝐶))
4140notbid 307 . . . . . . 7 (𝑦 = 𝐵 → (¬ 𝑦𝑅𝐶 ↔ ¬ 𝐵𝑅𝐶))
4241rspcv 3437 . . . . . 6 (𝐵 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 → ¬ 𝐵𝑅𝐶))
4339, 42syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 → ¬ 𝐵𝑅𝐶))
44 snsstp2 4485 . . . . . . 7 {𝐶} ⊆ {𝐵, 𝐶, 𝐷}
45 snssg 4451 . . . . . . . 8 (𝐶𝐴 → (𝐶 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐶} ⊆ {𝐵, 𝐶, 𝐷}))
466, 45syl 17 . . . . . . 7 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐶 ∈ {𝐵, 𝐶, 𝐷} ↔ {𝐶} ⊆ {𝐵, 𝐶, 𝐷}))
4744, 46mpbiri 248 . . . . . 6 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → 𝐶 ∈ {𝐵, 𝐶, 𝐷})
48 breq1 4799 . . . . . . . 8 (𝑦 = 𝐶 → (𝑦𝑅𝐷𝐶𝑅𝐷))
4948notbid 307 . . . . . . 7 (𝑦 = 𝐶 → (¬ 𝑦𝑅𝐷 ↔ ¬ 𝐶𝑅𝐷))
5049rspcv 3437 . . . . . 6 (𝐶 ∈ {𝐵, 𝐶, 𝐷} → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷 → ¬ 𝐶𝑅𝐷))
5147, 50syl 17 . . . . 5 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷 → ¬ 𝐶𝑅𝐷))
5235, 43, 513orim123d 1548 . . . 4 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐵 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐶 ∨ ∀𝑦 ∈ {𝐵, 𝐶, 𝐷} ¬ 𝑦𝑅𝐷) → (¬ 𝐷𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐷)))
5327, 52mpd 15 . . 3 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (¬ 𝐷𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐷))
54 3ianor 1096 . . 3 (¬ (𝐷𝑅𝐵𝐵𝑅𝐶𝐶𝑅𝐷) ↔ (¬ 𝐷𝑅𝐵 ∨ ¬ 𝐵𝑅𝐶 ∨ ¬ 𝐶𝑅𝐷))
5553, 54sylibr 224 . 2 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐷𝑅𝐵𝐵𝑅𝐶𝐶𝑅𝐷))
56 3anrot 1087 . 2 ((𝐷𝑅𝐵𝐵𝑅𝐶𝐶𝑅𝐷) ↔ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))
5755, 56sylnib 317 1 ((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3o 1071  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wral 3042  wrex 3043  Vcvv 3332  cun 3705  wss 3707  c0 4050  {csn 4313  {cpr 4315  {ctp 4317   class class class wbr 4796   Fr wfr 5214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-br 4797  df-fr 5217
This theorem is referenced by:  epne3  7137  dfwe2  7138
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